# Categorical centers and Yetter–Drinfel‘d-modules as 2-categorical (bi)lax
structures

Bojana Femić
Mathematical Institute of
Serbian Academy of Sciences and Arts
Kneza Mihaila 36,
11 000 Belgrade, Serbia
<EMAIL_ADDRESS>Corresponding author Sebastian Halbig
Research Group Algebraic Lie Theory
Philipps-Universität Marburg
Biegenstraße 10,
35037 Marburg, Germany
<EMAIL_ADDRESS>
###### Abstract

The bicategorical point of view provides a natural setting for many concepts
in the representation theory of monoidal categories. We show that centers of
twisted bimodule categories correspond to categories of 2-dimensional natural
transformations and modifications between the deloopings of the twisting
functors. We also show that dualities lift to centers of twisted bimodule
categories. Inspired by the notion of (pre)bimonoidal functors due to McCurdy
and Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensional
functors which are simultaneously lax and colax with a compatibility
condition. Our approach uses a sort of 2-categorical Yang-Baxter operators,
but the idea could equally be carried out using a kind of 2-categorical
braidings. We show how this concept, which we call bilax functors, generalize
many known notions from the theory of Hopf algebras. We propose a 2-category
of bilax functors whose 1-cells generalize the notions of Yetter-Drinfel‘d
modules in ordinary categories, and a type of bimonads and mixed distributive
laws in 2-categories. We show that the 2-category of bilax functors from the
trivial 2-category is isomorphic to the 2-category of bimonads, and that there
is a faithful 2-functor from the latter to the 2-category of mixed
distributive laws of Power and Watanabe.

Keywords: center categories, bicategories, Yang-Baxter operators, bimonads,
bimonoidal functors.

2020 MSC: 18N10, 18D25, 18M15.

##  1 Introduction

The concept of centers of a monoids was categorified independently by
Drinfel‘d, Majid and Street in the 1990’s. Since then it has been extensively
studied in Hopf algebra and category theory, see for example [17] for an
overview. One of its striking features comes from the fact that by passing
from sets to categories one can replace the qualitative question: ‘Do two
elements commute with another?’ with a quantitative one: ‘How many suitably
coherent (iso-)morphisms exist between the tensor product of two objects and
its opposite?’. Such (iso-)morphisms are called _half-braidings_. The center
of a monoidal category consists of objects of the underlying category with
fixed half-braidings together with morphisms of the base category which
satisfy a certain compatibility relation.

The aim of the present paper is twofold. In the first part, we study the
center construction from the bicategorical perspective. Our main motivation
comes from the observation that monoidal categories can be identified with
bicategories with a single object. This procedure, sometimes called delooping,
establishes an equivalence between monoidal categories with monoidal functors
and bicategories with a single object together with pseudofunctors. Following
this line of thinking, by simple means we reveal a surprising and beautiful
fact that colax natural transformations between lax functors among
bicategories with single objects are nothing but the objects of the twisted
Drinfel‘d center of the corresponding codomain monoidal category. Accordingly,
modifications of such colax transformations correspond to the morphisms in the
Drinfel‘d center, so that one has an isomorphism of categories. In particular,
we obtain:

###### Theorem 1.1

Let ${\mathcal{C}}$ be a monoidal category and write $Del({\mathcal{C}})$ for
its delooping. There exists a monoidal equivalence of categories between the
Drinfel‘d center of ${\mathcal{C}}$ and the category of pseudonatural
transformations and their modifications on the identitity functor of
$Del({\mathcal{C}})$.

In fact, we formulate a general (weak) center category
${\mathcal{Z}}^{w}(F,{\mathcal{M}},G)$ for a
${\mathcal{D}}\mbox{-}{\mathcal{C}}$-bimodule category ${\mathcal{M}}$ and two
lax monoidal functors $F:{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ and
$G:{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{C}}$ from a third monoidal
category ${\mathcal{E}}$. In the case that
${\mathcal{C}}={\mathcal{D}}={\mathcal{M}}$, we interpret
${\mathcal{Z}}^{w}(F,{\mathcal{D}},G)$ from a bicategorical point of view. As
this interpretation relies on delooping, in paricular on the fact that the
monoidal product of ${\mathcal{C}}$ becomes the composition of 1-cells, this
approach does not allow for giving a bicategorical interpretation of
${\mathcal{Z}}^{w}(F,{\mathcal{M}},G)$ for general $F$ and $G$. This task will
be treated elsewhere. (The weakness corresponds to dealing with non-invertible
half-braidings, while with strongness we allude to invertible ones.
Accordingly, we differentiate left and right weak centers. )

We prove that left weak center categories form a bicategory
${\mathcal{Z}}^{w}_{l}({\mathcal{E}},{\mathcal{K}})$ which is isomorphic to
the bicategory of suitable lax functors of bicategories. When ${\mathcal{K}}$
is an autonomous 2-category, meaning that all its 1-cells have left and right
adjoints, and ${\mathcal{E}}$ is an autonomous monoidal category, then on
pseudofunctors $Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ the weak and strong
center categories coincide (Proposition 3.14). Furthermore, under these
conditions the corresponding bicategory
${\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{E}},{\mathcal{K}})$ is autonomous
(Proposition 3.15). This result provides a natural interpretation of duality
notions between centers of compatible bimodule categories in [14]. Moreover,
our bicategorical interpretation of center categories
${\mathcal{Z}}^{w}_{l}(F,{\mathcal{M}},G)$ which make up the bicategory
${\mathcal{Z}}^{w}_{l}({\mathcal{E}},{\mathcal{K}})$ encompasses also the
Shimizu’s bicategory ${\rm TF}({\mathcal{C}},{\mathcal{D}})$ of tensor
functors from [29] and the result thereof about duals.

The above bicategory ${\mathcal{Z}}^{w}_{l}({\mathcal{E}},{\mathcal{K}})$ is a
particular case of the bicategory
$\operatorname{Lax}_{clx}({\mathcal{B}},{\mathcal{B}}^{\prime})$ of lax
functors ${\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$ among
bicategories, colax natural transformations and modifications. Taking the
pseudo-pseudo version of the latter, we get the bicategory ${\rm
Ps}_{ps}({\mathcal{B}},{\mathcal{B}}^{\prime})$ whose hom-categories are ${\rm
Ps}({\mathcal{F}},{\mathcal{G}})$ for pseudofunctors
${\mathcal{F}},{\mathcal{G}}:{\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$. When
${\mathcal{F}}={\mathcal{G}}=\operatorname{Id}_{\mathcal{B}}$ one recovers the
center category ${\mathcal{Z}}({\mathcal{B}})$ of the bicategory
${\mathcal{B}}$ introduced in [25]. Our above bicategory of center categories
alludes to the possibility to consider “twisted center categories of the
bicategory ${\mathcal{B}}^{\prime}$”.

On the other hand, in the second part of the paper, we introduce and study
2-categorical functors which are simultaneously lax and colax with a
compatibility relation involving a Yang–Baxter operator. We call them bilax
functors and differentiate bilax functors with compatible Yang-Baxter
operator. For monoidal categories such functors were studied under the name of
pre-bimonoidal and bimonoidal functors in [24], and (when the domain category
is braided) bilax functors in [1]. We show that our bilax functors generalize
a variety of notions and possess certain preservation properties: bialgebras
in braided monoidal categories, bimonads in 2-categories (with respect to
Yang-Baxter operators, YBO’s), and preserve bimonads (w.r.t. YBO’s), bimonads
in 2-categories with respect to distributive laws from [10], module comonads
and comodule monads, and relative bimonad modules. Moreover, the component
functors of a bilax functor on hom-categories factor through the category of
Hopf bimodules (w.r.t. YBO’s). The 2-categorical notions in italic letters are
introduced in this paper and they generalize to 2-categories the same named
notions in braided monoidal categories.

We record that instead of working with Yang-Baxter operators, one could
equally use local braidings, following the footsteps of [1]. In this case the
generalization and preservation results somewhat differ from the ones that we
obtained and that are listed above.

We establish a 2-category of bilax functors ${\rm
Bilax}({\mathcal{K}},{\mathcal{K}}^{\prime})$ by introducing bilax natural
transformations and bilax modifications. Accordingly, ${\rm
Bilax}_{c}({\mathcal{K}},{\mathcal{K}}^{\prime})$ denotes the 2-category of
bilax functors with compatible Yang-Baxter operator. Bilax natural
transformations are both lax and colax natural transformations satisfying a
compatibility condition. As such they generalize bimonad morphisms from [10]
and Yetter-Drinfel‘d modules from braided monoidal categories. In the
classical case, the category of Yetter-Drinfel‘d modules over a bialgebra $B$
is monoidally equivalent to the Drinfel‘d center of the category
${}_{B}{\mathcal{M}}$ of modules over the same bialgebra. The half-braidings
in the left Drinfel‘d center can be seen as colax natural transformations. In
the category ${}_{B}{\mathcal{M}}$ one can construct a lax natural
transformation which together with the colax one makes a bilax natural
transformation. (We explain this in more detail at the end of Subsection 5.1.)
This illustrates why in a general 2-category ${\mathcal{K}}$ bilax natural
transformations (and bilax modifications) generalize the category of Yetter-
Drinfel‘d modules, but not the (left) center category. Finally, we show that
there is a 2-category isomorphism ${\rm Bilax}_{c}(1,{\mathcal{K}})\cong{\rm
Bimnd}({\mathcal{K}})$ and a faithful 2-functor ${\rm
Bimnd}({\mathcal{K}})\hookrightarrow\operatorname{Dist}({\mathcal{K}})$. Here
${\rm Bimnd}({\mathcal{K}})$ is the 2-category of bimonads from [11] and
$\operatorname{Dist}({\mathcal{K}})$ is the 2-category of mixed distributive
laws of [28].

The paper is composed as follows. We first give an overview of bicategories,
deloopings, module and center categories. In section 3 we give a higher
categorical interpretation of center categories and study when the bicategory
of center categories is autonomous. Bilax functors and their properties are
studied in section 4, while in the last section a 2-category of bilax functors
is introduced and its relations to the 2-categories ${\rm
Bimnd}({\mathcal{K}})$ and $\operatorname{Dist}({\mathcal{K}})$ is shown.

##  2 Preliminaries: Deloopings and weak twisted centers

We assume that the reader is familiar with the notion of a braided monoidal
category and the corresponding notation of string diagrams (see e.g. [15, 17,
31]), as well as with the definition of a bicategory, for which we recommend
[3, 16].

In this section we give a short summary of bicategories, delooping
bicategories, module categories and weak twisted centers. For a more extensive
discussion of module categories we refer the reader to [7].

Briefly, a _monoidal category_ consists of a category ${\mathcal{C}}$ together
with a suitably associative and unital multiplication
$\otimes\colon{\mathcal{C}}\times{\mathcal{C}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{C}}$ implemented by a functor
which is called the _tensor product_.

A ‘many object’ generalization of monoidal categories is provided by
_bicategories_. These can be thought of as higher dimensional categories with
_hom-categories_ between every pair of objects instead of mere sets. The
objects of these hom-categories are called _1-cells_ and the morphisms
_2-cells_. Any bicategory ${\mathcal{K}}$ admits two ways to compose:
_horizontal composition_ given by the composition functors

$\circ_{Z,Y,X}\colon{\mathcal{K}}(Y,Z)\times{\mathcal{K}}(X,Y)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}(X,Z),\qquad\qquad\text{
for }X,Y,Z\in{\mathcal{O}b}{\mathcal{K}}\text{ (objects of ${\mathcal{K}}$)},$

and _vertical composition_ induced by the compositions inside the hom-
categories. Instead of identity morphisms, every
$X\in{\mathcal{O}b}{\mathcal{K}}$ has a unit 1-cell
$\operatorname{id}_{X}\in{\mathcal{K}}(X,X)$. In general, the horizontal
composition of a bicategory is associative and unital only up to suitable
natural isomorphisms. Bicategories where these morphisms are identities are
called _2-categories_. Since every bicategory is biequivalent to a 2-category,
we will restrict ourselves without loss of generality to the setting of
2-categories.

As hinted at before, there is an intimate relationship between monoidal
categories and bicategories. It is provided by considering a monoidal category
${\mathcal{C}}$ as a bicategory $Del({\mathcal{C}})$ with one object (which we
will usually denote by $*$) and ${\mathcal{C}}$ as its unique hom-category.
Under this identification, the tensor product of ${\mathcal{C}}$ becomes the
horizontal composition of $Del({\mathcal{C}})$, and the monoidal unit becomes
the identity 1-cell $id$ on the unique object of $Del({\mathcal{C}})$. The
resulting canoncial isomorphism of categories between the category of monoidal
categories with certain structure preserving functors and one-object
bicategories plus structure preserving 2-dimensional functors is called
_delooping_ :

$Del\colon\left\\{\begin{gathered}\text{monoidal categories with}\\\
\text{lax/colax/strong monoidal
functors}\end{gathered}\right\\}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}\left\\{\begin{gathered}\text{one-object
bicategories with}\\\ \text{lax/colax/pseudofunctors}.\end{gathered}\right\\}$
(1)

###### Remark 2.1

Observe that we consider the horizontal composition in bicategories in the
counter lexicographical order, whereas the tensor product in a monoidal
category ${\mathcal{C}}$ is read from left to right, that is:
${\mathcal{C}}\times{\mathcal{C}}\ni(X,Y)\stackrel{{\scriptstyle\otimes}}{{\mapsto}}X\otimes
Y\in{\mathcal{C}}$. For objects $X,Y\in{\mathcal{C}}$ corresponding to 1-cells
$x,y$ in $Del({\mathcal{C}})$ respectively, this implies that the tensor
product $X\otimes Y$ corresponds to the composition of 1-cells $y\circ x$. In
order to avoid applying this mirror symmetry, we are going to consider in the
isomorphism $Del$ in (1) that the reversed tensor product becomes the
horizontal composition in bicategories (formally, this is precomposing $Del$
with the isomorphism functor defined on objects by sending a monoidal category
$({\mathcal{C}},\otimes)$ to it reversed
category$({\mathcal{C}}^{rev},\otimes^{rev})$ ).

Bicategories provide a natural interpretation of the representation theory of
monoidal categories. All endomorphism categories of a bicategory are monoidal
with horizontal composition as a tensor product. Similarly, given two objects
$A,B\in{\mathcal{O}b}{\mathcal{K}}$ of a bicategory with endomorphism
categories ${\mathcal{D}}:={\mathcal{K}}(A,A)$ and
${\mathcal{C}}:={\mathcal{K}}(B,B)$, horizontal composition endows
${\mathcal{M}}\coloneqq{\mathcal{K}}(A,B)$ with the structure of a
$({\mathcal{C}},{\mathcal{D}})$-_bimodule category_. That is, there are two
functors
$\rhd\colon{\mathcal{C}}\times{\mathcal{M}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{M}}$ and
$\lhd\colon{\mathcal{M}}\times{\mathcal{D}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ subject to analogous but
weakened version of the axioms of bimodules over a monoid.

Conversely: to any $({\mathcal{C}},{\mathcal{D}})$-bimodule category
${\mathcal{M}}$ we can associate a two object bicategory $Del({\mathcal{M}})$,
which we call the _delooping_ of ${\mathcal{M}}$. It has two objects $0$ and
$1$ and hom-categories
$Del({\mathcal{M}})(0,0)={\mathcal{D}},Del({\mathcal{M}})(0,1)=M,Del({\mathcal{M}})(1,1)={\mathcal{C}}$
and $Del({\mathcal{M}})(1,0)=1$, the trivial category. Horizontal composition
is given by the tensor products of ${\mathcal{C}}$ and ${\mathcal{D}}$ and the
left and right action of ${\mathcal{C}}$ and ${\mathcal{D}}$ on
${\mathcal{M}}$. The relation between (bi)module categories and bicategories
was already observed by Benaboú, [3, Section 2.3].

If the categories ${\mathcal{C}}$ and ${\mathcal{D}}$ coincide, one can define
the center of a bimodule category. The aim of the paper at hand will be the
study of these centers and their interaction with the theory of bicategories
in a slightly more general version.

###### Definition 2.2

Let $F\colon{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ and
$G\colon{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{C}}$ be lax monoidal functors
and ${\mathcal{M}}$ a (strict) $({\mathcal{C}},{\mathcal{D}})$-bimodule
category over the (strict) monoidal categories ${\mathcal{C}}$ and
${\mathcal{D}}$. A _left half-braiding_ of an object $M\in{\mathcal{M}}$
relative to $F$ and $G$ is a natural transformation

$\sigma_{X}\colon M\lhd F(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}G(X)\rhd M,\qquad\qquad\text{for all
}X\in{\mathcal{E}},$

such that for all $X,Y\in\mathcal{E}$ the following diagrams commute:

${{M\lhd F(Y)\lhd F(X)}}$${{G(Y)\rhd M\lhd F(X)}}$${{M\lhd F(Y\otimes
X)}}$${{G(Y\otimes X)\rhd M}}$${{G(Y)\rhd G(X)\rhd
M}}$$\scriptstyle{\sigma_{Y}\lhd\operatorname{id}_{F(X)}}$$\scriptstyle{\operatorname{id}_{G(Y)}\lhd\sigma_{X}}$$\scriptstyle{G^{2}\rhd
M}$$\scriptstyle{M\lhd F^{2}}$$\scriptstyle{\sigma_{Y\otimes X}}$ (2) ${{M\lhd
I\cong I\rhd M}}$${{G(I)\rhd M}}$${{M\lhd F(I)}}$$\scriptstyle{G^{0}\rhd
M}$$\scriptstyle{M\lhd F^{0}}$$\scriptstyle{\sigma_{I}}$ (3)

Similarly, a _right half-braiding_ on $M$ relative to $F$ and $G$ is a natural
transformation

$\tilde{\sigma}_{X}\colon G(X)\rhd M\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}M\lhd F(X),\qquad\qquad\text{for all
}X\in{\mathcal{E}},$

subject to analogous identities.

The _left weak center of ${\mathcal{M}}$ relative to $F$ and $G$_ is the
category ${\mathcal{Z}}_{l}^{w}(F,{\mathcal{M}},G)$. Its objects are pairs
$(M,\sigma)$ consisting of an object $M\in{\mathcal{M}}$ together with a left
half-braiding $\sigma$ on $M$ relative to $F$ and $G$. A morphism between
objects $(M,\sigma),(N,\tau)\in{\mathcal{Z}}_{l}^{w}(F,{\mathcal{M}},G)$ is an
arrow $f\in{\mathcal{M}}(M,N)$ such that

$(\operatorname{id}_{G(X)}\rhd
f)\sigma_{X}=\tau_{X}(f\lhd\operatorname{id}_{F(X)}),\qquad\qquad\text{for all
}X\in{\mathcal{E}}.$ (4)

The full subcategory ${\mathcal{Z}}^{s}_{l}(F,{\mathcal{M}},G)$ of
${\mathcal{Z}}_{l}^{w}(F,{\mathcal{M}},G)$ whose objects have invertible half-
braidings is called the _(strong) left center of ${\mathcal{M}}$ relative to
$F$ and $G$_. When the functors are clear from the context, we will call the
latter two categories simply left weak/strong twisted centers of
${\mathcal{M}}$, respectively.

We define the right weak and strong twisted center categories
${\mathcal{Z}}_{r}^{w}(F,{\mathcal{M}},G)$ and
${\mathcal{Z}}^{s}_{r}(F,{\mathcal{M}},G)$ in an analogous way.

When ${\mathcal{C}}={\mathcal{D}}={\mathcal{M}}$ we set
$Z_{l}^{w}(F,G):={\mathcal{Z}}_{l}^{w}(F,{\mathcal{D}},G)$ and
$Z_{r}^{w}(F,G):={\mathcal{Z}}_{r}^{w}(F,{\mathcal{D}},G)$. For
${\mathcal{C}}$ a tensor category and $F,G$ tensor functors these present the
(left and right version of) twisted center category $Z(F,G)$ studied in [29,
Section 3].

In case $F=G=\operatorname{Id}_{\mathcal{C}}$, we write
${\mathcal{Z}}^{l}_{\mathcal{C}}({\mathcal{M}})\coloneqq{\mathcal{Z}}^{s}_{l}(\operatorname{Id},{\mathcal{M}},\operatorname{Id})$
and
${\mathcal{Z}}^{r}_{\mathcal{C}}({\mathcal{M}})\coloneqq{\mathcal{Z}}^{s}_{r}(\operatorname{Id},{\mathcal{M}},\operatorname{Id})$.
These recover the (left and right) center category from [12]. If moreover
${\mathcal{M}}={\mathcal{C}}$, the categories
${\mathcal{Z}}^{l}_{\mathcal{C}}({\mathcal{C}})$ and
${\mathcal{Z}}^{r}_{\mathcal{C}}({\mathcal{C}})$ recover the left and right
Drinfel‘d center categories of ${\mathcal{C}}$.

###### Lemma 2.3

Suppose $G\colon{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{C}}$ and
$F\colon{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ are lax monoidal functors
and ${\mathcal{M}}$ is a $({\mathcal{C}},{\mathcal{D}})$-bimodule category.
Then there exists an isomorphism of categories

$\Xi\colon{\mathcal{Z}}_{l}^{s}(F,{\mathcal{M}},G)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{Z}}_{r}^{s}(F,{\mathcal{M}},G),\qquad\qquad\Xi(M,\sigma)=(M,\sigma^{-1}),$
(5)

which is the identity on morphisms.

###### Proof.

Suppose that $\sigma$ is an invertible left half-braiding on an object
$M\in{\mathcal{M}}$. We show that $\sigma^{-1}$ defines a right half-braiding.
Precomposing the equation in (2) by
$(\sigma_{Y}^{-1}\lhd\operatorname{id}_{F(X)})(\operatorname{id}_{G(X)}\rhd\sigma_{X}^{-1})$
and postcomposing by $\sigma_{YX}^{-1}$ yields the desired compatibility of
right half-braidings with the lax functor structures. Analogous calculations
show that $\sigma^{-1}$ is compatible with the lax units of $F$ and $G$ and
that $\Xi$ sends any morphism in the strong left center to a morphism of the
strong right center. The proof is concluded by constructing $\Xi^{-1}$ in the
same spirit as $\Xi$. That is, by mapping invertible right half-braidings to
their inverses. ∎∎

The construction of (left) strong twisted center categories can be seen as a
result of the following composition of 2-functors:

${\mathcal{C}}\mbox{-}{\mathcal{D}}\mbox{-}\operatorname{Bimod}\stackrel{{\scriptstyle(F,G)}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}{\mathcal{E}}\mbox{-}{\mathcal{E}}\mbox{-}\operatorname{Bimod}\stackrel{{\scriptstyle{\mathcal{Z}}_{\mathcal{E}}}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}{\mathcal{Z}}({\mathcal{E}})\mbox{-}\operatorname{Mod}$
$\hskip 28.45274pt{\mathcal{M}}\hskip 22.76228pt\mapsto\hskip
17.07182pt{}_{G}{\mathcal{M}}_{F}\hskip 22.76228pt\mapsto\hskip
11.38092pt{\mathcal{Z}}_{\mathcal{E}}({}_{G}{\mathcal{M}}_{F})$

where $(F,G)$ denotes precomposing the left and right action by $F$ and $G$,
respectively, and ${\mathcal{Z}}_{\mathcal{E}}$ is defined as in [8, Section
3.4]. The term “twisted” is motivated by this composition. Namely, if $F$ and
$G$ are strong monoidal functors, a
${\mathcal{C}}\mbox{-}{\mathcal{D}}$-bimodule category structure is twisted by
them into an ${\mathcal{E}}$-bimodule structure.

##  3 Categorical centers as a data in a tricategory

At the core of our investigation in this section are (weak) twisted centers
and their interpretation from a higher categorical point of view. We will show
that center categories are hom-categories of hom-bicategories of a particular
tricategory. Namely, the tricategory of bicategories with a single object.

###  3.1 Categorical centers as (co)lax natural transformations

For the interpretation of center categories from the perspective of
2-categories we first recall the definitions of lax and colax functors between
bicategories and of lax and colax natural transformations between the latter.

###### Definition 3.1

A _lax functor_
$({\mathcal{F}},{\mathcal{F}}^{2},{\mathcal{F}}^{0})\colon{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ between
2-categories consists of

  1. 1.

an assignment ${\mathcal{O}b}{\mathcal{K}}\ni
A\mapsto{\mathcal{F}}(A)\in{\mathcal{O}b}{\mathcal{K}}^{\prime}$,

  2. 2.

for all $A,B\in{\mathcal{O}b}{\mathcal{K}}$ a local functor
${\mathcal{F}}_{A,B}\colon{\mathcal{K}}(A,B)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}({\mathcal{F}}(A),{\mathcal{F}}(B))$,

  3. 3.

a natural transformation

${\mathcal{F}}^{2}_{g,f}\colon F(g)\otimes^{\prime}F(f)\Rightarrow F(g\otimes
f),\qquad\qquad\text{ for
}(g,f)\in{\mathcal{K}}(B,C)\times{\mathcal{K}}(A,B),$

and

  4. 4.

a natural transformation

${\mathcal{F}}^{0}_{A}\colon
id_{{\mathcal{F}}(A)}\Rightarrow{\mathcal{F}}(id_{A}),\qquad\qquad\text{ for
}A\in{\mathcal{O}b}{\mathcal{K}},$

so that ${\mathcal{F}}^{2}$ and ${\mathcal{F}}^{0}$ satisfy _associativity_
and _unitality_ laws.
When the natural transformations ${\mathcal{F}}^{2}$ and ${\mathcal{F}}^{0}$
are directed in the opposite direction and satisfy _coassociativity_ and
_counitality_ laws, one has a colax functor. One speaks about a pseudofunctor
if ${\mathcal{F}}^{2}$ and ${\mathcal{F}}^{0}$ are isomorphisms.

Lax transformations can be defined both for lax and colax functors. The same
holds for colax transformations, so that there are four variations of
definitions, depending on the situation.

###### Definition 3.2

Let
$({\mathcal{F}},{\mathcal{F}}^{2},{\mathcal{F}}^{0})\colon{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ and
$({\mathcal{G}},{\mathcal{G}}^{2},{\mathcal{G}}^{0})\colon{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ be lax functors
between 2-categories. A colax natural transformation
$\chi\colon{\mathcal{F}}\Rightarrow{\mathcal{G}}$ consists of

  1. 1.

a 1-cell $\chi_{A}\colon{\mathcal{F}}(A)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{G}}(A)$ for each object
$A\in{\mathcal{O}b}{\mathcal{K}}$, and

  2. 2.

for every pair of objects $A,B\in Ob{\mathcal{K}}$ a collection of 2-cells

$\\{\chi_{f}\colon\chi_{B}\circ{\mathcal{F}}_{A,B}(f)\Rightarrow{\mathcal{G}}_{A,B}(f)\circ\chi_{A}\mid
f\in{\mathcal{K}}(A,B)\\}$ (6)

natural in $f$ subject to _colax multiplicativity_

${{{\mathcal{F}}(B)}}$${{{\mathcal{F}}(A)}}$${{{\mathcal{F}}(B)}}$${{{\mathcal{F}}(C)}}$${{{\mathcal{F}}(A)}}$${{{\mathcal{F}}(C)}}$${{=}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{G}}(B)}}$${{{\mathcal{G}}(C)}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{G}}(C)}}$$\scriptstyle{{\mathcal{F}}(f)}$$\scriptstyle{{\mathcal{F}}(g)}$$\scriptstyle{\chi_{C}}$$\scriptstyle{{\mathcal{G}}(g)}$$\scriptstyle{\chi_{B}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{{\mathcal{G}}(f)}$$\scriptstyle{{\mathcal{G}}(gf)}$$\scriptstyle{{\mathcal{F}}(f)}$$\scriptstyle{{\mathcal{F}}(g)}$$\scriptstyle{{\mathcal{F}}(gf)}$$\scriptstyle{{\mathcal{G}}(gf)}$$\scriptstyle{\chi_{C}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{\chi_{f}}$$\scriptstyle{\chi_{g}}$$\scriptstyle{\chi_{gf}}$$\scriptstyle{{\mathcal{F}}^{2}}$$\scriptstyle{{\mathcal{G}}^{2}}$
(7)

and _colax unitality_

${{{\mathcal{F}}(A)}}$${{{\mathcal{F}}(A)}}$${{{\mathcal{F}}(A)}}$${{{\mathcal{F}}(A)}}$${{=}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{G}}(A)}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{\operatorname{id}_{{\mathcal{G}}(A)}}$$\scriptstyle{{\mathcal{G}}(\operatorname{id}_{A})}$$\scriptstyle{\operatorname{id}_{{\mathcal{F}}(A)}}$$\scriptstyle{\operatorname{id}_{{\mathcal{F}}(A)}}$$\scriptstyle{{\mathcal{F}}(\operatorname{id}_{A})}$$\scriptstyle{\chi_{A}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{{\mathcal{G}}(\operatorname{id}_{A})}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\chi_{\operatorname{id}_{A}}}$$\scriptstyle{{\mathcal{F}}^{0}}$$\scriptstyle{{\mathcal{G}}^{0}}$
(8)

If the 2-cells of $\chi$ are invertible, it is called a _pseudonatural
transformation_. In case they are identities, one speaks of a _strict natural
transformation_.

By reverting the direction of the 2-cells of $\chi$ one obtains the notion of
a lax natural transformation between lax functors.

We start by a simple observation that entails a marvelous fact.

###### Proposition 3.3

Let $F,G\colon{\mathcal{C}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ be lax monoidal functors.
The objects of the weak twisted center
${\mathcal{Z}}_{l}^{w}(F,{\mathcal{D}},G)$ are canonically in bijection with
colax natural transformations $\chi\colon Del(F)\Rightarrow Del(G)$ between
the induced lax functors
$Del(F),Del(G):Del({\mathcal{C}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}Del({\mathcal{D}})$. Under this
identification, the objects of the strong center correspond to pseudonatural
transformations.

###### Proof.

Since both bicategories $Del({\mathcal{C}}),Del({\mathcal{D}})$ have a single
object, there is a single 1-cell component of $\chi\colon Del(F)\Rightarrow
Del(G)$, which is a distinguished object $D_{\chi}=\chi_{*}$ in
${\mathcal{D}}$. The 2-cell components of $\chi$ amount to morphisms
$\chi_{X}:D_{\chi}\otimes F(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}G(X)\otimes D_{\chi}$ in ${\mathcal{D}}$
natural in $X$ (mind that we do not flip the order of factors when translating
from $Del({\mathcal{C}})$ to ${\mathcal{D}}$, as we assume the reversed tensor
product in the sense of Remark 2.1), and the colax multiplicativity and unity
translate into the commuting diagrams (2) and (3). The second claim is
immediate. ∎∎

###### Remark 3.4

An analogous statement to the previous proposition for right half-braidings
can be obtained by considering lax instead of colax natural transformations
between lax functors.

To obtain a bicategorical interpretation of the morphisms in a center
category, we need to recall the definition of modifications.

###### Definition 3.5

A modification $a:\chi\Rrightarrow\psi$ between two colax natural
transformations
$\chi,\psi:{\mathcal{F}}\Rightarrow{\mathcal{G}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ consists of a
family of 2-cells $a_{A}:\chi(A)\Rightarrow\psi(A)$, indexed by the objects
$A\in{\mathcal{O}b}{\mathcal{K}}$, such that for every 1-cell
$f\in{\mathcal{K}}(A,B)$ we have:

${{{\mathcal{F}}(A)}}$${{{\mathcal{G}}(A)}}$${{{\mathcal{F}}(A)}}$${{{\mathcal{G}}(A)}}$${{=}}$${{{\mathcal{F}}(B)}}$${{{\mathcal{G}}(B)}}$${{{\mathcal{F}}(B)}}$${{{\mathcal{G}}(B)}}$$\scriptstyle{\chi_{A}}$$\scriptstyle{F(f)}$$\scriptstyle{G(f)}$$\scriptstyle{\chi_{B}}$$\scriptstyle{\psi_{A}}$$\scriptstyle{\chi_{f}}$$\scriptstyle{\psi_{A}}$$\scriptstyle{G(f)}$$\scriptstyle{F(f)}$$\scriptstyle{\psi_{B}}$$\scriptstyle{\chi_{B}}$$\scriptstyle{\psi_{f}}$$\scriptstyle{a_{A}}$$\scriptstyle{a_{B}}$
(9)

For two lax functors
${\mathcal{F}},{\mathcal{G}}:{\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$ among
bicategories let $\text{Colax}({\mathcal{F}},{\mathcal{G}})$ and
$\text{Lax}({\mathcal{F}},{\mathcal{G}})$ denote the categories of colax
(respectively lax) natural transformations and their modifications. Similarly,
$\text{Pseudo}(F,G)$ denotes the category of pseudonatural transformations and
their modifications.

###### Proposition 3.6

Let $F,G\colon{\mathcal{C}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ be lax monoidal functors.
There are canonical isomorphisms of categories:

${\mathcal{Z}}_{l}^{w}(F,{\mathcal{D}},G)\cong\text{Colax}(Del(F),Del(G)),\qquad\qquad{\mathcal{Z}}_{l}^{s}(F,{\mathcal{D}},G)\cong\text{Pseudo}(Del(F),Del(G)),$
${\mathcal{Z}}_{r}^{w}(F,{\mathcal{D}},G)\cong\text{Lax}(Del(F),Del(G)),\qquad\qquad{\mathcal{Z}}_{r}^{s}(F,{\mathcal{D}},G)\cong\text{Pseudo}(Del(F),Del(G)).$

###### Proof.

We only prove the claims for left center categories, as the other cases are
analogous. Let $\chi,\psi\colon Del(F)\Rightarrow Del(G)$ be colax natural
transformations and write
$(D_{\chi},\chi),(E_{\psi},\psi)\in{\mathcal{Z}}_{w}^{l}(F,{\mathcal{D}},G)$
for their corresponding objects in the weak left center. Since
$Del({\mathcal{C}})$ and $Del({\mathcal{C}})$ are deloopings of monoidal
categories, any modification $a\colon\chi\Rrightarrow\psi$ is defined by a
single morphism $f\colon D_{\chi}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}E_{\psi}$ satisfying for all
$X\in{\mathcal{C}}$ the following identity:

$\begin{minipage}{24.0pt}
\begin{picture}(24.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle D$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(X)$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(18.0,60.0){\line(0,-1){12.0}}
\put(6.0,48.0){\line(0,-1){1.0}} \put(6.0,36.0){\line(0,1){1.0}}
\put(1.0,37.0){\line(1,0){12.0}} \put(1.0,47.0){\line(1,0){12.0}}
\put(1.0,47.0){\line(0,-1){10.0}}
\put(18.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}$}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(23.0,37.0){\line(-1,0){12.0}} \put(23.0,47.0){\line(-1,0){12.0}}
\put(23.0,47.0){\line(0,-1){10.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\framebox(10.0,10.0)[]{$\scriptstyle f$}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle E$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{24.0pt}
\begin{picture}(24.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle D$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(X)$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(18.0,60.0){\line(0,-1){12.0}}
\put(6.0,48.0){\line(0,-1){1.0}} \put(6.0,36.0){\line(0,1){1.0}}
\put(1.0,37.0){\framebox(10.0,10.0)[]{$\scriptstyle f$}}
\put(18.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{X}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle E.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (10)

This is precisely the defining equation of a morphism in the weak center and
the claim follows. ∎∎

In the above proposition we started from two lax monoidal functors between
monoidal categories to obtain the result. One can also start from two lax
functors between 2-categories in a specific way to obtain an analogous result.
For this purpose recall the interplay between ordinary categories and
bicategories encoded in the delooping isomorphism (1) and the discussion after
Remark 2.1.

###### Proposition 3.7

Let ${\mathcal{E}}$ be a monoidal category, ${\mathcal{K}}$ a bicategory which
has at least two objects $0$ and $1$, and assume that
${\mathcal{F}},{\mathcal{G}}:Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ are two lax functors such
that ${\mathcal{F}}(*)=0$ and ${\mathcal{G}}(*)=1$. There are canonical
isomorphisms of categories

${\mathcal{Z}}_{l}^{w}(F,{\mathcal{M}},G)\cong\text{Colax}({\mathcal{F}},{\mathcal{G}}),\qquad\qquad{\mathcal{Z}}_{l}^{s}(F,{\mathcal{M}},G)\cong\text{Pseudo}({\mathcal{F}},{\mathcal{G}}),$
${\mathcal{Z}}_{r}^{w}(F,{\mathcal{M}},G)\cong\text{Lax}({\mathcal{F}},{\mathcal{G}}),\qquad\qquad{\mathcal{Z}}_{r}^{s}(F,{\mathcal{M}},G)\cong\text{Pseudo}({\mathcal{F}},{\mathcal{G}}),$

for a suitable bimodule category ${\mathcal{M}}$ and lax monoidal functors $F$
and $G$.

###### Proof.

We set ${\mathcal{D}}:={\mathcal{K}}(0,0),{\mathcal{C}}:={\mathcal{K}}(1,1)$
and ${\mathcal{M}}:={\mathcal{K}}(0,1)$. Then the two lax functors yield lax
monoidal functors
$F={\mathcal{F}}_{*,*}:{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}},G={\mathcal{G}}_{*,*}:{\mathcal{E}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{C}}$. The unique 1-cell
component of a colax natural transformation
$\chi:{\mathcal{F}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{G}}$ is an object $M$ living
in the $({\mathcal{C}}\mbox{-}{\mathcal{D}})$-bimodule category
${\mathcal{M}}$, as it is a 1-cell mapping
$0={\mathcal{F}}(*)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{G}}(*)=1$ in ${\mathcal{K}}$.
The 2-cell component yields a half-braiding given by morphisms $\chi_{X}:M\lhd
F(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}G(X)\rhd M$ in ${\mathcal{M}}$, for
every 1-endocell in ${\mathcal{K}}$, i.e. an object $X\in{\mathcal{E}}$. The
rest follows as in Proposition 3.6. ∎∎

Recall that we recovered the left center category from [12] as
${\mathcal{Z}}^{l}_{\mathcal{C}}({\mathcal{M}})\coloneqq{\mathcal{Z}}^{s}_{l}(\operatorname{Id},{\mathcal{M}},\operatorname{Id})$,
and similarly the right one, and that they are isomorphic by Lemma 2.3.

###### Corollary 3.8

Let ${\mathcal{C}}$ be a monoidal category, ${\mathcal{K}}$ a bicategory which
has at least two objects $0$ and $1$, and assume that
${\mathcal{F}},{\mathcal{G}}:Del({\mathcal{C}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ are two pseudofunctors
such that ${\mathcal{F}}(*)=0,{\mathcal{G}}(*)=1$ and
${\mathcal{F}}_{*,*}={\mathcal{G}}_{*,*}=\operatorname{Id}_{\mathcal{C}}$.
There are canonical isomorphisms of categories

${\mathcal{Z}}^{l}_{\mathcal{C}}({\mathcal{M}})\cong{\rm
Pseudo}({\mathcal{F}},{\mathcal{G}})\cong{\mathcal{Z}}^{r}_{\mathcal{C}}({\mathcal{M}})$

where ${\mathcal{M}}={\mathcal{K}}(0,1)$.

###  3.2 The bicategory of center categories

The bicategorical perspective gives us a deeper insight of “why” the twisted
center categories can be composed between each other, and in particular “why”
“$F\mbox{-}F$-twisted” center categories are monoidal. Namely, for fixed
bicategories ${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$ there are
bicategories $\operatorname{Lax}_{lx}({\mathcal{B}},{\mathcal{B}}^{\prime})$
and $\operatorname{Lax}_{clx}({\mathcal{B}},{\mathcal{B}}^{\prime})$ of lax
functors ${\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$, lax (resp.
colax) transformations and their modifications, see [16, Theorem 4.4.11]. The
composition of (co)lax natural transformations (as 1-cells in these
bicategories) is given by what is known as the vertical composition of such
transformations. The hom-categories are
$\text{Lax}({\mathcal{F}},{\mathcal{G}})$ and
$\text{Colax}({\mathcal{F}},{\mathcal{G}})$ for two lax functors
${\mathcal{F}},{\mathcal{G}}$, respectively. The horizontal composition in the
two bicategories corresponds to the composition of the right, respectively
left, weak twisted center categories between themselves. In particular, the
fact that $\operatorname{Lax}({\mathcal{F}},{\mathcal{F}})$ and
$\text{Colax}({\mathcal{F}},{\mathcal{F}})$ are monoidal entails that the
“$F\mbox{-}F$-twisted” center categories are monoidal as well. We explain now
how this is achieved.

Let us fix a monoidal category ${\mathcal{E}}$ and a 2-category
${\mathcal{K}}$ of which we think as a collection of monoidal categories and
composable bimodule categories between them. According to Proposition 3.7, we
can identify the hom-categories of
$\operatorname{Lax}_{clx}(Del({\mathcal{E}}),{\mathcal{K}})$ with the weak
left centers relative to lax monoidal functors from ${\mathcal{E}}$ to the
endo-1-categories of ${\mathcal{K}}$. In order to emphasize this point of
view, we write
${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})\coloneqq\operatorname{Lax}_{clx}(Del({\mathcal{E}}),{\mathcal{K}})$.
Given lax functors ${\mathcal{F}},{\mathcal{G}}\colon
Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$, we write
${\mathcal{Z}}_{l}^{w}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})$ for the
hom-category of ${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})$ with
source ${\mathcal{F}}$ and target ${\mathcal{G}}$. (Observe that they are
categories ${\mathcal{Z}}_{l}^{w}(F,{\mathcal{M}},G)$ from Proposition 3.7,
where $F,G$ are lax monoidal functors induced by ${\mathcal{F}},{\mathcal{G}}$
and ${\mathcal{M}}={\mathcal{K}}(0,1)$.)

The above observation readily implies that (weak left) centers can be
organized into a bicategory. We refer to
${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})$ as the _bicategory of
weak left centers of ${\mathcal{K}}$ relative to ${\mathcal{E}}$_. Our next
result translates its horizontal composition into a ‘pre-tensor product’
between weak relative centers. Given two vertically composable 2-cells
$x\stackrel{{\scriptstyle\alpha}}{{\Rightarrow}}y\stackrel{{\scriptstyle\beta}}{{\Rightarrow}}z:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B$, we denote their vertical composition
in equations by the fraction $\frac{\alpha}{\beta}$.

###### Proposition 3.9

Let ${\mathcal{E}}$ be a monoidal category and ${\mathcal{K}}$ a 2-category.
The weak left centers of ${\mathcal{K}}$ relative to ${\mathcal{E}}$ form a
bicategory ${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})$. For all lax
functors
${\mathcal{F}},{\mathcal{G}},{\mathcal{H}}:Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$, its horizontal
composition is given by

$\displaystyle\circ_{{\mathcal{F}},{\mathcal{G}},{\mathcal{H}}}\colon{\mathcal{Z}}_{l}^{w}({\mathcal{G}},{\mathcal{K}},{\mathcal{H}})\times{\mathcal{Z}}_{l}^{w}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})$
$\displaystyle\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{Z}}_{l}^{w}({\mathcal{F}},{\mathcal{K}},{\mathcal{H}})$
(11) $\displaystyle\left((N,\tau),(M,\sigma)\right)$
$\displaystyle\mapsto(N\circ
M,\frac{\operatorname{id}_{N}\circ\sigma}{\tau\circ\operatorname{id}_{M}})$
$\displaystyle(g,f)$ $\displaystyle\mapsto(g\circ f)$

###### Proof.

The first claim is merely a recapitulation that
${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})=\operatorname{Lax}_{clx}(Del({\mathcal{E}}),{\mathcal{K}})$
is a bicategory. By Proposition 3.7, the objects and morphisms of any weak
center are in correspondence with appropriate colax natural transformations
and modifications. Applying this identification, one immediately obtains the
above formulas from their respective composition rules. (Observe that
$M\in{}_{\mathcal{C}}{\mathcal{M}}_{\mathcal{D}}={\mathcal{K}}(0,1)$ and
$N\in{}_{\mathcal{P}}{\mathcal{N}}_{\mathcal{C}}={\mathcal{K}}(1,2)$, where
${\mathcal{F}}(*)=0,{\mathcal{G}}(*)=1$ and ${\mathcal{H}}(*)=2$, and
${\mathcal{D}}:={\mathcal{K}}(0,0),{\mathcal{C}}:={\mathcal{K}}(1,1)$ and
${\mathcal{P}}:={\mathcal{K}}(2,2)$, in the sense of the proof of Proposition
3.7.) ∎∎

Analogous considerations hold for the bicategory of weak right centers
${\mathcal{Z}}_{r}^{w}({\mathcal{E}},{\mathcal{K}})\coloneqq\operatorname{Lax}_{lx}(Del({\mathcal{E}}),{\mathcal{K}})$.

The consequences of the above result are best exemplified by considering the
bicategory ${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{E}})$ of all twisted
weak left centers ${\mathcal{Z}}_{l}^{w}(F,{\mathcal{E}},G)$ of
${\mathcal{E}}$. The weak left analogue ${\mathcal{Z}}_{l}^{w}({\mathcal{E}})$
of the Drinfel‘d center corresponds to the endo-category on the identity
functor of ${\mathcal{E}}$. The previous result recovers its monoidal
structure. Moreover, it implies that, if only the left or right action of
${\mathcal{E}}$ on itself is twisted, the resulting center canonically becomes
a right, respectively, left module category over
${\mathcal{Z}}_{l}^{w}({\mathcal{E}})$. This gives a theoretical justification
to the constructions of [14] involving the anti-Drinfel‘d center and its
opposite.

Let us consider the pseudo-pseudo version of the bicategories
$\operatorname{Lax}_{lx}({\mathcal{B}},{\mathcal{B}}^{\prime})$ and
$\operatorname{Lax}_{clx}({\mathcal{B}},{\mathcal{B}}^{\prime})$. It is a
bicategory that we denote by ${\rm
Ps}_{ps}({\mathcal{B}},{\mathcal{B}}^{\prime})$ with hom-categories ${\rm
Ps}({\mathcal{F}},{\mathcal{G}})$ for pseudofunctors
${\mathcal{F}},{\mathcal{G}}:{\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$. In the
particular case when
${\mathcal{F}}={\mathcal{G}}=\operatorname{Id}_{\mathcal{B}}$ we have the
center category ${\mathcal{Z}}({\mathcal{B}})$ of the bicategory
${\mathcal{B}}$ introduced in [25]. Our above considerations allude to the
possibility to consider “twisted center categories of the bicategory
${\mathcal{B}}^{\prime}$”. In this case the twisting is done through the
pseudo- (or lax or colax) functors
${\mathcal{B}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{B}}^{\prime}$.

###  3.3 The tricategory that encomapsses strong center categories

Let ${\rm Bicat}$ denote the tricategory of bicategories, pseudofunctors,
pseudonatural transformations, and modifications.

###### Remark 3.10

Although it is sufficient to consider only lax functors and colax (resp. lax)
natural transformations in order to recover left (resp. right) weak twisted
center category, in order to form a tricategory whose 2- and 3-cells are some
kind of bicategorical functors and transformations, both kinds of cells should
be of pseudo type, i.e. they both should have isomorphisms for their
respective defining structures. Namely, on one hand, in order to be able to
define the horizontal composition of two lax (or colax) natural
transformations, both lax and colax structures of the functors they act to are
needed (see e.g. Second problem in [18]). On the other hand, one sees from the
diagram (11.3.12) of [16, Chapter 11.3] that in order to construct the
isomorphism interchange 3-cell one needs both pseudofunctors and pseudonatural
transformations.

One clearly has:

###### Proposition 3.11

By delooping, the full sub-tricategory of ${\rm Bicat}$ whose objects are
bicategories with a single object gives rise to a tricategory ${\rm
Bicat}^{*}$ with monoidal categories as objects, strong monoidal functors as
1-cells, and for each pair
$F,G\colon{\mathcal{C}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ of such functors the
strong center
${\mathcal{Z}}_{l}^{s}(F,{\mathcal{D}},G)\cong{\mathcal{Z}}_{r}^{s}(F,{\mathcal{D}},G)$
as hom-category.

In view of Remark 3.10 it becomes clear why we do not speak of a tricategory
that contains weak center categories as bottom hom-categories.

By further restricting the tricategory ${\rm Bicat}^{*}$ to finite tensor
categories (in the sense of [7]), one has that every hom-bicategory ${\rm
Bicat}^{*}({\mathcal{C}},{\mathcal{D}})$ is precisely the bicategory ${\rm
TF}({\mathcal{C}},{\mathcal{D}})$ from [29, Section 3.2], of tensor functors
between finite tensor categories ${\mathcal{C}}$ and ${\mathcal{D}}$. As the
author works in a context of rigid categories, he shows that for every hom-
category $Z_{l}^{w}(F,G)$ of the bicategory ${\rm
TF}({\mathcal{C}},{\mathcal{D}})$ and every object $(V,\sigma_{V})\in
Z_{l}^{w}(F,G)$ the transformation $\sigma_{V}$ is invertible ([29, Lemma
3.1]), and $(V,\sigma_{V})$ has a left and a right dual object in
$Z_{l}^{w}(G,F)$. We will generalize this to 2-categories in Proposition 3.14.

Question. In Proposition 3.7 we dealt with lax functors
${\mathcal{F}},{\mathcal{G}}:Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ from a one-
object bicategory. If we allow for the domain bicategory ${\mathcal{K}}$ to
have more than one object, we wonder what the categories of (co)lax
transformations $\chi:{\mathcal{F}}\Rightarrow{\mathcal{G}}$ recover. In
particular, what is the “meaning in nature” of the 2-cell components
$\chi_{X}:\chi(B)\circ F(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}G(X)\circ\chi(A)$ for 1-cells
$X:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}B$
in ${\mathcal{K}}$?

###  3.4 String diagrams in 2-categories

In Subsection 3.5 and throughout Section 4 and Section 5 we will use string
diagrams for 2-categories (again relying on the biequivalence of any
bicategory with a 2-category). Our string diagrams are read from top to bottom
and (in the context of 2-categories) from right to left. The domains and
codomains of the strings stand for 1-cells, while the strings themselves and
boxes stand for 2-cells. The 0-cells are to be understood from the context
(reading the 1-cells from right to left). Observe that such string diagrams
which depict 2-cells in a 2-category ${\mathcal{K}}$ acting on the same
underlying 0-cells $A\in{\mathcal{O}b}{\mathcal{K}}$ (that is, morphisms in
the monoidal categories ${\mathcal{K}}(A,A)$ for every $A$) correspond exactly
to the string diagrams in the monoidal categories ${\mathcal{K}}(A,A)$ (due to
the isomorphism (1)). This is even more clear if
${\mathcal{K}}=Del({\mathcal{C}})$ for some monoidal category ${\mathcal{C}}$.

Let ${\mathcal{F}}$ be a lax functor and ${\mathcal{G}}$ a colax functor. We
depict their lax, respectively colax structures by diagrams in the following
way:

$\scriptstyle{\mathcal{F}}(g)$$\scriptstyle{\mathcal{F}}(f)$$\scriptstyle{\mathcal{F}}(gf)$
$\scriptstyle id_{{\mathcal{F}}(A)}$$\scriptstyle{\mathcal{F}}(id_{A})$
$\scriptstyle{\mathcal{G}}(gf)$$\scriptstyle{\mathcal{G}}(g)$$\scriptstyle{\mathcal{G}}(f)$
$\scriptstyle{\mathcal{G}}(id_{A})$$\scriptstyle id_{{\mathcal{F}}(A)}$

where $g,f$ are composable 1-cells and $A$ a 0-cell in the domain 2-category.
We will often simplify the notation $\circ$ for the composition of 1-cells by
concatenation.

Observe that a colax transformation between two lax functors, Definition 3.2,
is nothing but a distributive law between lax functor structures that is
moreover natural in 1-cells. Similarly, a colax transformation between two
colax functors is a distributive law between colax functor structures that is
moreover natural in 1-cells. In string diagrams we may write the latter as
follows:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{C}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gf)$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(24.0,36.0){\oval(12.0,18.0)[t]}
\put(24.0,48.0){\line(0,-1){3.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{g}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(30.0,36.0){\line(0,-1){12.0}}
\qbezier(6.0,24.0)(6.0,20.0)(0.0,18.0)\qbezier(0.0,18.0)(-6.0,16.0)(-6.0,12.0)
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(13.0,13.0){\line(1,0){12.0}} \put(13.0,23.0){\line(1,0){12.0}}
\put(13.0,23.0){\line(0,-1){10.0}}
\put(30.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{f}$}}
\put(30.0,24.0){\line(0,-1){1.0}} \put(30.0,12.0){\line(0,1){1.0}}
\put(35.0,13.0){\line(-1,0){12.0}} \put(35.0,23.0){\line(-1,0){12.0}}
\put(35.0,23.0){\line(0,-1){10.0}}
\put(-6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(g)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(f)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{C}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gf)$}}
\qbezier(12.0,48.0)(12.0,44.0)(12.0,42.0)\qbezier(12.0,42.0)(12.0,40.0)(12.0,36.0)
\qbezier(24.0,48.0)(24.0,44.0)(24.0,42.0)\qbezier(24.0,42.0)(24.0,40.0)(24.0,36.0)
\put(18.0,36.0){\line(0,-1){1.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,25.0){\line(0,1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{gf}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(24.0,12.0){\oval(12.0,18.0)[t]}
\put(24.0,24.0){\line(0,-1){3.0}}
\qbezier(36.0,24.0)(36.0,20.0)(39.0,18.0)\qbezier(39.0,18.0)(42.0,16.0)(42.0,12.0)
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(g)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(f)$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage};\quad\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id_{A})$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(6.0,36.0){\line(0,-1){1.0}} \put(6.0,24.0){\line(0,1){1.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{id_{A}}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){6.0}}
\put(6.0,18.0){\circle{5.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id_{A})$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(6.0,36.0){\line(0,-1){24.0}} \put(18.0,36.0){\line(0,-1){6.0}}
\put(18.0,30.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage};\quad\begin{minipage}{36.0pt}
\begin{picture}(36.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{B}$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(18.0,60.0){\line(0,-1){12.0}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,48.0){\line(0,-1){1.0}}
\put(18.0,36.0){\line(0,1){1.0}} \put(13.0,37.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(1,0){12.0}} \put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\alpha)$}}
\put(35.0,47.0){\line(-1,0){12.0}} \put(35.0,37.0){\line(-1,0){12.0}}
\put(35.0,37.0){\line(0,1){10.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{y}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(y)$}}
\put(24.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{24.0pt}
\begin{picture}(24.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{B}$}}
\put(30.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(18.0,60.0){\line(0,-1){12.0}} \put(30.0,60.0){\line(0,-1){12.0}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(13.0,37.0){\line(1,0){12.0}} \put(13.0,47.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{x}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{G}}(\alpha)$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(30.0,36.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){12.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(y)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (12)

for any 2-cell $\alpha:x\Rightarrow y:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B$. (A colax transformation between two
lax functors is determined by string diagrams that are both vertically and
horizontally symmetric to the above ones.) Dually (taking opposite 2-cells,
i.e. taking vertically symmetric diagrams to those in (12)) we obtain string
diagrams for lax transformations between two lax functors.

###  3.5 Adjoints in 2-categories

Dualisability plays a prominent role in the study of monoidal categories and
the closely related subject of (extended) topological quantum field theories,
see [2]. For example, Section 2 of [6] and Section 4 of [14] discuss and
utilize ‘duals’ of bimodule categories and centers. In the following, we want
to provide a 2-categorical perspective on these constructions, thereby giving
a theoretical underpinning for some of the ad-hoc constructions of [14].

We start by briefly recalling the notion of adjoint 1-cells in 2-categories,
see for example [13, 19]. Let $f\colon A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B$ be a 1-cell in a 2-category
${\mathcal{K}}$. A _left adjoint_ of $f$ is a 1-cell $u\colon
B\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
together with two 2-cells
$\eta:\operatorname{id}_{A}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}uf$ and
$\varepsilon:fu\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}\operatorname{id}_{B}$ such that

$\frac{\operatorname{Id}_{f}\circ\eta}{\varepsilon\circ\operatorname{Id}_{f}}=\operatorname{id}_{f}\qquad\qquad\text{and}\qquad\qquad\frac{\eta\circ\operatorname{Id}_{u}}{\operatorname{Id}_{u}\circ\varepsilon}=\operatorname{id}_{u}.$

Similarly, a _right adjoint_ of $f$ is a 1-cell $v\colon
B\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
together with two 2-cells
$\bar{\eta}\colon\operatorname{id}_{B}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}fv$ and $\bar{\varepsilon}\colon
vf\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}\operatorname{id}_{A}$ such that

$\frac{\operatorname{Id}_{v}\circ\bar{\eta}}{\bar{\varepsilon}\circ\operatorname{Id}_{v}}=\operatorname{id}_{f}\qquad\qquad\text{and}\qquad\qquad\frac{\bar{\eta}\circ\operatorname{Id}_{f}}{f\circ\bar{\varepsilon}}=\operatorname{id}_{f}.$

In string diagrams we will write $\eta=\begin{minipage}{24.0pt}
\begin{picture}(24.0,12.0)\put(0.0,0.0){\line(1,0){0.0}}
\put(12.0,0.0){\oval(12.0,18.0)[t]} \put(0.0,12.0){\line(1,0){0.0}}
\end{picture} {\vskip 9.68745pt} \end{minipage}$ and
$\varepsilon=\begin{minipage}{24.0pt}
\begin{picture}(24.0,12.0)\put(0.0,0.0){\line(1,0){0.0}}
\put(12.0,12.0){\oval(12.0,18.0)[b]} \put(0.0,12.0){\line(1,0){0.0}}
\end{picture} {\vskip 9.68745pt} \end{minipage}$, and they satisfy the laws:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(30.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle u$}}
\put(12.0,24.0){\oval(12.0,18.0)[t]} \put(30.0,36.0){\line(0,-1){12.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(24.0,24.0){\oval(12.0,18.0)[b]}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle u$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\operatorname{Id}_{u}\qquad\textnormal{and}\qquad\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle f$}}
\put(18.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(24.0,24.0){\oval(12.0,18.0)[t]}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(30.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle f$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\operatorname{Id}_{f}.$

Pseudofunctors ${\mathcal{F}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ preserve (and
reflect) adjoints and we have:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,24.0)\put(0.0,12.0){\line(1,0){0.0}}
\put(6.0,16.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}}
\put(30.0,16.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(u)$}}
\put(18.0,12.0){\oval(24.0,18.0)[b]} \put(0.0,12.0){\line(1,0){0.0}}
\end{picture} {\vskip 9.68745pt} \end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}}
\put(30.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(u)$}}
\put(18.0,36.0){\oval(24.0,18.0)[b]} \put(18.0,27.0){\line(0,-1){3.0}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,13.0){\line(0,1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\varepsilon)$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(23.0,13.0){\line(-1,0){12.0}} \put(23.0,23.0){\line(-1,0){12.0}}
\put(23.0,23.0){\line(0,-1){10.0}} \put(18.0,12.0){\line(0,-1){6.0}}
\put(18.0,6.0){\circle{5.0}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture}
{\vskip 9.68745pt} \end{minipage},\qquad\begin{minipage}{36.0pt}
\begin{picture}(36.0,24.0)\put(0.0,12.0){\line(1,0){0.0}}
\put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(u)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(18.0,36.0){\line(0,1){6.0}} \put(18.0,42.0){\circle{5.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\eta)$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(u)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f).$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (13)

In the aforementioned [14], certain trace-like morphisms were considered in
order to implement pivotal structures on the Drinfel‘d centers. This involves
a ‘lift’ of the notion of duals, i.e. adjoints, to the setting of centers of
bimodule categories. With our interpretation of
${\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})=\operatorname{Lax}_{clx}(Del({\mathcal{E}}),{\mathcal{K}})$
in Proposition 3.9 as a bicategory, for a monoidal category ${\mathcal{E}}$
and a bicategory ${\mathcal{K}}$, we can now derive a more conceptual version
of this construction.

###### Definition 3.12

We refer to a 2-category ${\mathcal{K}}$ as _autonomous_ if all 1-cells in
${\mathcal{K}}$ have left and right adjoints.

The most prominent example of an autonomous 2-category is given by the
delooping $Del({\mathcal{E}})$ of an autonomous (rigid) monoidal category
${\mathcal{E}}$. We will prove at the end of this section that the full sub-
bicategory ${\mathcal{Z}}_{l}^{w-ps}({\mathcal{E}},{\mathcal{K}})$ whose
objects are pseudofunctors is autonomous. We will use the following fact,
which is immediately proved:

###### Lemma 3.13

Let
${\mathcal{F}},{\mathcal{G}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ be
pseudofunctors between 2-categories, and suppose that
$\chi:{\mathcal{F}}\Rightarrow{\mathcal{G}}$ is a colax natural transformation
with respect to their lax functors structures. Then $\chi$ is a colax natural
transformation of pseudofunctors.

The following result is a 2-categorical interpretation of the fact that the
half-braidings over autonomous monoidal categories are automatically
invertible, see [29, Lemma 3.1].

###### Proposition 3.14

Let ${\mathcal{K}}$ be an autonomous 2-category and ${\mathcal{E}}$ an
autonomous monoidal category. For any pair of pseudofunctors
${\mathcal{F}},{\mathcal{G}}\colon
Del({\mathcal{E}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$, the weak and strong
centers coincide. Moreover, the inverse of a left half-braiding is a right
half-braiding, that is:
${\mathcal{Z}}_{l}^{w-ps}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})={\mathcal{Z}}_{l}^{s-ps}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})\cong{\mathcal{Z}}_{r}^{s-ps}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})={\mathcal{Z}}_{r}^{w-ps}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})$.

###### Proof.

For the first claim it suffices to show that the component 2-cells of any
colax natural transformation between pseudofunctors
$\chi\colon{\mathcal{F}}\Rightarrow{\mathcal{G}}$ are invertible. Hereto, we
fix an object $X\in{\mathcal{E}}$ and write $X^{\ast}$ for its left adjoint.
Starting from the left hand-side below and applying the left equality in (13),
the coherence of $\chi$ with the functor’s multiplicativity, naturality of
$\chi$ with respect to $\varepsilon_{X}$ and the coherence of $\chi$ with
respect to the functor’s counitalities (mind Lemma 3.13) we reach the equality
with the right hand-side:

$\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}$}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\hskip
5.69046pt{\mathcal{F}}(X)$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X^{*})$}}
\put(6.0,48.0){\line(0,-1){1.0}} \put(6.0,36.0){\line(0,1){1.0}}
\put(1.0,37.0){\line(1,0){12.0}} \put(1.0,47.0){\line(1,0){12.0}}
\put(1.0,47.0){\line(0,-1){10.0}}
\put(18.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}$}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(23.0,37.0){\line(-1,0){12.0}} \put(23.0,47.0){\line(-1,0){12.0}}
\put(23.0,47.0){\line(0,-1){10.0}} \put(30.0,48.0){\line(0,-1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X^{*}}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(30.0,24.0){\line(0,-1){12.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(48.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X^{*})$}}
\put(6.0,48.0){\line(0,-1){36.0}} \put(30.0,48.0){\oval(24.0,18.0)[b]}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

Then the 2-cell:

$\gamma_{X}:=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(X)$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}$}}
\put(6.0,48.0){\line(0,-1){24.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(36.0,36.0){\oval(12.0,18.0)[t]} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X^{*}}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){24.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(30.0,24.0){\line(0,-1){12.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle D_{\chi}$}}
\put(48.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

is clearly a left inverse of
$\chi_{X}:D_{\chi}\circ{\mathcal{F}}(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{G}}(X)\circ D_{\chi}$. The
analogous reasoning, but this time using the coherence of $\chi$ with the
comultiplicativity and unitality of the functors, shows that $\gamma_{X}$ is
also a right inverse of $\chi_{X}$. The last statement is proved directly. ∎∎

Our next result states that adjoints can be ‘lifted’ to weak centers.

###### Proposition 3.15

Suppose ${\mathcal{E}}$ to be an autonomous monoidal category and
${\mathcal{K}}$ to be an autonomous 2-category. Then the full sub-bicategory
${\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{E}},{\mathcal{K}})\subset{\mathcal{Z}}_{l}^{w}({\mathcal{E}},{\mathcal{K}})$
whose objects are pseudofunctors is autonomous.

###### Proof.

We only show that any 1-cell in
${\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{E}},{\mathcal{K}})$, that is, an
object
$(M,\chi)\in{\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{F}},{\mathcal{K}},{\mathcal{G}})$,
has a right adjoint living in
${\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{G}},{\mathcal{K}},{\mathcal{F}})$.
The case of left adjoints is analogous. Due to Proposition 3.14, the half-
braiding $\chi$, i.e. the 2-cells
$\chi_{X}:M\circ{\mathcal{F}}(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{G}}(X)\circ M$, is invertible.
We utilize this fact, to define a half-braiding on ${}^{\ast}M$ as shown in
the next diagram

$\sigma_{X}:=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(X)$}}
\put(6.0,48.0){\line(0,-1){24.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(36.0,36.0){\oval(12.0,18.0)[t]} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}^{-1}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){24.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(30.0,24.0){\line(0,-1){12.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(48.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

A direct computation shows that it satisfies the Equations (2) and (3). For
example, we have

$\begin{minipage}{84.0pt}
\begin{picture}(84.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(0.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(Y)$}}
\put(54.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(X)$}}
\put(6.0,72.0){\line(0,-1){24.0}} \put(18.0,72.0){\line(0,-1){12.0}}
\put(36.0,60.0){\oval(12.0,18.0)[t]} \put(54.0,72.0){\line(0,-1){24.0}}
\put(18.0,60.0){\line(0,-1){1.0}} \put(18.0,48.0){\line(0,1){1.0}}
\put(13.0,49.0){\line(1,0){12.0}} \put(13.0,59.0){\line(1,0){12.0}}
\put(13.0,59.0){\line(0,-1){10.0}}
\put(30.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{Y}^{-1}$}}
\put(30.0,60.0){\line(0,-1){1.0}} \put(30.0,48.0){\line(0,1){1.0}}
\put(35.0,49.0){\line(-1,0){12.0}} \put(35.0,59.0){\line(-1,0){12.0}}
\put(35.0,59.0){\line(0,-1){10.0}} \put(42.0,60.0){\line(0,-1){24.0}}
\put(72.0,48.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\oval(12.0,18.0)[b]}
\put(30.0,48.0){\line(0,-1){24.0}} \put(54.0,48.0){\line(0,-1){1.0}}
\put(54.0,36.0){\line(0,1){1.0}} \put(49.0,37.0){\line(1,0){12.0}}
\put(49.0,47.0){\line(1,0){12.0}} \put(49.0,47.0){\line(0,-1){10.0}}
\put(66.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}^{-1}$}}
\put(66.0,48.0){\line(0,-1){1.0}} \put(66.0,36.0){\line(0,1){1.0}}
\put(71.0,37.0){\line(-1,0){12.0}} \put(71.0,47.0){\line(-1,0){12.0}}
\put(71.0,47.0){\line(0,-1){10.0}} \put(78.0,48.0){\line(0,-1){36.0}}
\put(48.0,36.0){\oval(12.0,18.0)[b]} \put(66.0,36.0){\line(0,-1){12.0}}
\put(48.0,24.0){\oval(36.0,18.0)[b]} \put(48.0,15.0){\line(0,-1){3.0}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(Y\otimes X)$}}
\put(78.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{60.0pt}
\begin{picture}(60.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(0.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(Y)$}}
\put(36.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(X)$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(18.0,60.0){\line(0,-1){12.0}}
\put(30.0,60.0){\line(0,-1){12.0}} \put(48.0,48.0){\oval(12.0,18.0)[t]}
\put(6.0,48.0){\line(0,-1){24.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(25.0,37.0){\line(1,0){12.0}} \put(25.0,47.0){\line(1,0){12.0}}
\put(25.0,47.0){\line(0,-1){10.0}}
\put(42.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}^{-1}$}}
\put(42.0,48.0){\line(0,-1){1.0}} \put(42.0,36.0){\line(0,1){1.0}}
\put(47.0,37.0){\line(-1,0){12.0}} \put(47.0,47.0){\line(-1,0){12.0}}
\put(47.0,47.0){\line(0,-1){10.0}} \put(54.0,48.0){\line(0,-1){24.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{Y}^{-1}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(36.0,24.0){\oval(12.0,18.0)[b]}
\put(36.0,15.0){\line(0,-2){3.0}} \put(54.0,24.0){\line(0,-1){12.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(Y\otimes X)$}}
\put(54.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{72.0pt}
\begin{picture}(72.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(0.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(Y)$}}
\put(36.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(X)$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(30.0,60.0){\oval(24.0,18.0)[b]}
\put(30.0,51.0){\line(0,-1){3.0}}
\qbezier(6.0,48.0)(6.0,40.0)(12.0,36.0)\qbezier(12.0,36.0)(18.0,32.0)(18.0,24.0)
\put(30.0,48.0){\line(0,-1){12.0}} \put(48.0,36.0){\oval(12.0,18.0)[t]}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(25.0,25.0){\line(1,0){12.0}} \put(25.0,35.0){\line(1,0){12.0}}
\put(25.0,35.0){\line(0,-1){10.0}}
\put(42.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{Y\otimes X}^{-1}$}}
\put(42.0,36.0){\line(0,-1){1.0}} \put(42.0,24.0){\line(0,1){1.0}}
\put(47.0,25.0){\line(-1,0){12.0}} \put(47.0,35.0){\line(-1,0){12.0}}
\put(47.0,35.0){\line(0,-1){10.0}} \put(54.0,36.0){\line(0,-1){24.0}}
\put(24.0,24.0){\oval(12.0,18.0)[b]} \put(42.0,24.0){\line(0,-1){12.0}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(Y\otimes X)$}}
\put(60.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

To conclude the proof, we show that the unit
$\eta\colon\operatorname{id}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}M\circ{}^{\ast}M$ and counit
$\varepsilon\colon{}^{\ast}M\circ M\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}\operatorname{id}$ lift to morphisms in
the center category
${\mathcal{Z}}^{w\mbox{-}ps}_{l}({\mathcal{G}},{\mathcal{K}},{\mathcal{F}})$.
For the counit this follows from the computation depicted below.

$\begin{minipage}{60.0pt}
\begin{picture}(60.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(0.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(36.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(6.0,72.0){\line(0,-1){48.0}} \put(18.0,72.0){\line(0,-1){1.0}}
\put(18.0,60.0){\line(0,1){1.0}} \put(13.0,61.0){\line(1,0){12.0}}
\put(13.0,71.0){\line(1,0){12.0}} \put(13.0,71.0){\line(0,-1){10.0}}
\put(30.0,66.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}$}}
\put(30.0,72.0){\line(0,-1){1.0}} \put(30.0,60.0){\line(0,1){1.0}}
\put(35.0,61.0){\line(-1,0){12.0}} \put(35.0,71.0){\line(-1,0){12.0}}
\put(35.0,71.0){\line(0,-1){10.0}} \put(18.0,60.0){\line(0,-1){24.0}}
\qbezier(30.0,60.0)(30.0,56.0)(42.0,54.0)\qbezier(42.0,54.0)(54.0,52.0)(54.0,48.0)
\put(36.0,36.0){\oval(12.0,18.0)[t]} \put(54.0,48.0){\line(0,-1){24.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}^{-1}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(30.0,24.0){\line(0,-1){12.0}}
\put(48.0,24.0){\oval(12.0,18.0)[b]}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(6.0,48.0){\line(0,-1){24.0}} \put(18.0,48.0){\line(0,-1){1.0}}
\put(18.0,36.0){\line(0,1){1.0}} \put(13.0,37.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(1,0){12.0}} \put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\chi_{X}^{-1}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(30.0,24.0){\line(0,-1){12.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{}^{\ast}M$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(12.0,48.0){\oval(12.0,18.0)[b]} \put(30.0,48.0){\line(0,-1){36.0}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(X)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

An analogous argument shows that the unit also becomes a morphism in the
center. ∎

##  4 Bilax functors

We are interested in functors on bicategories that are both lax and colax but
not necessarily pseudofunctors. Likewise, we are interested in natural
transformations that are both lax and colax but not necessarily pseudonatural
transformations, as well as in their modifications. In particular, we
introduce the notions of a bilax functor, bilax natural transformation and
bilax modifications. We formulate them for 2-categories, just to avoid the use
of associators and unitors, but the corresponding definitions for bicategories
can be formulated in a straightforward fashion. To that end, we fix
2-categories ${\mathcal{K}}$ and ${\mathcal{K}}^{\prime}$. In this section we
introduce bilax functors, and leave the resting two notions for the next
section.

###  4.1 Bilax functors

We reiterate that we will often simplify the notation $\circ$ for the
horizontal composition of 1- and 2-cells by concatenation.

###### Definition 4.1

Let ${\mathcal{F}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ be a 2-functor.
A Yang-Baxter operator for ${\mathcal{F}}$ consists of a collection of 2-cells
$\nu_{g,f}:{\mathcal{F}}(g){\mathcal{F}}(f)\Rightarrow{\mathcal{F}}(f){\mathcal{F}}(g)$
in ${\mathcal{K}}^{\prime}$, natural in 1-endocells $f,g$ of ${\mathcal{K}}$,
which satisfy the Yang-Baxter equation

$\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(h)$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(g)$}}
\put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}}
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for all 1-endocells $f,g,h$, and the following unity-counity law

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\end{minipage}$ (15)

We call a Yang-Baxter operator for the identity 2-functor
$\operatorname{Id}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ a Yang-Baxter operator of
${\mathcal{K}}$. We reserve the notation $c$ for a Yang-Baxter operator on the
identity 2-functor on ${\mathcal{K}}$.

For 2-categories ${\mathcal{K}}=Del({\mathcal{C}})$ where ${\mathcal{C}}$ is a
braided monoidal category, a class of Yang-Baxter operators $c$ is given by
the braiding(s) of ${\mathcal{C}}$.

###### Definition 4.2

Assume that ${\mathcal{K}}$ possesses a Yang-Baxter operator $c$. A bilax
functor is a pair
$({\mathcal{F}},\nu):({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ where
${\mathcal{F}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ is
simultaneously a lax and a colax functor and $\nu$ is a Yang-Baxter operator
of ${\mathcal{F}}$, meaning that apart from the rule (14) two additional
groups of rules hold: left and right lax distributive laws

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\end{minipage}=\begin{minipage}{24.0pt}
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\end{minipage}$ (16)

and left and right colax distributive laws

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\begin{picture}(24.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
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\end{minipage},$ (17)

and additionally the bilaxity condition

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\put(6.0,6.0){\circle{5.0}} \put(18.0,12.0){\line(0,-1){6.0}}
\put(18.0,6.0){\circle{5.0}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture}
{\vskip 9.68745pt} \end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id_{A})$}}
\put(30.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id_{A})$}}
\put(18.0,24.0){\oval(24.0,18.0)[b]} \put(18.0,15.0){\line(0,-1){3.0}}
\put(18.0,12.0){\line(0,-1){6.0}} \put(18.0,6.0){\circle{5.0}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\quad\begin{minipage}{12.0pt}
\begin{picture}(12.0,24.0)\put(0.0,12.0){\line(1,0){0.0}}
\put(6.0,12.0){\line(0,1){6.0}} \put(6.0,18.0){\circle{5.0}}
\put(6.0,12.0){\line(0,-1){6.0}} \put(6.0,6.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\operatorname{Id}_{id_{A}}$ (18)

holds for 1-cells $A\stackrel{{\scriptstyle
k}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
h}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
f}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
g}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}C$.

Observe that the unit laws in (16) (or the counit laws in (17)) together with
the fourth rule in (18) imply (15). We record that the unit laws of (16) (or
the counit laws of (17)) imply

$\begin{minipage}{24.0pt}
\begin{picture}(24.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}}
\put(6.0,36.0){\line(0,1){6.0}} \put(6.0,42.0){\circle{5.0}}
\put(18.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{1,1}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){6.0}}
\put(6.0,18.0){\circle{5.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{12.0pt}
\begin{picture}(12.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}}
\put(6.0,36.0){\line(0,-1){6.0}} \put(6.0,30.0){\circle{5.0}}
\put(6.0,12.0){\line(0,1){6.0}} \put(6.0,18.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{24.0pt}
\begin{picture}(24.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,1){6.0}}
\put(18.0,42.0){\circle{5.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{1,1}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){6.0}} \put(18.0,18.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id).$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (19)

We briefly comment the term “distributive law” in equtions (16) and (17) (we
also used it in (12)). This term in the context of lax functors appeared in
[9, Definition 3.1], as the authors say: “by analogy to the distributive laws
of monads, which have similar axioms”. On the other hand, monads and lax
monoidal functors on (monoidal) categories can both be interpreted as monoids:
the former are monoids in endofunctor categories, while the latter are monoids
under Day convolution. In accordance with this suggestive similarity we use
the term “distributive law” also for pseudonatural transformations on
lax/colax/bilax functors.

Let $({\mathcal{K}},c)$ and $({\mathcal{K}}^{\prime},d)$ be 2-categories with
their respective Yang-Baxter operators, and assume that a bilax functor
$({\mathcal{F}},\nu)$ is given acting between them. If
$\nu_{f,g}=d_{{\mathcal{F}}(f),{\mathcal{F}}(g)}$ for all composable
1-endocells $f,g$ in ${\mathcal{K}}$, we will say that ${\mathcal{F}}$ is a
bilax functor with a compatible Yang-Baxter operator, and we will write simply
${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$. In the case
that the relation between $\nu$ and $d$ is not known, we will write
$({\mathcal{F}},\nu):({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$, as in
Definition 4.2.

Given a bilax functor $({\mathcal{F}},\nu)$ the functors on endo-hom-
categories

${\mathcal{F}}_{A,A}:{\mathcal{K}}(A,A)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}({\mathcal{F}}(A),{\mathcal{F}}(A))$
(20)

are pre-bimonoidal in the sense of [24, Section 2], where we rely on the
strictification theorem for monoidal categories. If instead of the Yang-Baxter
operators one works with braidings on the endo-hom categories, then the
functors ${\mathcal{F}}_{A,A}$ are bilax as in [1, Section 19.9.1]. The latter
inspired the terminology in Definition 4.2. While the analogues of the
coherence conditions (16) and (17) do not appear explicitly in [24, Section
2], we incorporate them in accordance with the discussion in [1, Section
19.9.1] in our definition. Accordingly, we recover a “braided bialgebra” from
[31, Definition 5.1], as we will show further below. In the situation
${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ (i.e. that
$\nu$ is compatible with $d$), the functors ${\mathcal{F}}_{A,A}$ are
bimonoidal in the sense of [24, Section 2]. The other way around, we clearly
have:

###### Example 4.3

Let $({\mathcal{C}},\Phi_{\mathcal{C}})$ and
$({\mathcal{D}},\Phi_{\mathcal{D}})$ be braided monoidal categories. We
identify their braidings with Yang-Baxter operators $c$ and $d$ on their
delooping categories $Del({\mathcal{C}})$ and $Del({\mathcal{D}})$,
respectively. Any bilax functor
${\mathcal{F}}:(Del({\mathcal{C}}),c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}(Del({\mathcal{C}}),d)$ with a
compatible Yang-Baxter operator is a bimonoidal functor in the sense of [24,
Section 2].

###### Example 4.4

Let $({\mathcal{C}},\Phi)$ be a braided monoidal category and $1$ the trivial
2-category (it has a single 0-cell and only identity higher cells). Any bilax
functor ${\mathcal{F}}:1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}Del({\mathcal{C}})$ with invertible
$\nu$ which coincided with $\Phi$ can be identified with a bialgebra in
${\mathcal{C}}$. To see this, note that ${\mathcal{F}}$ determines and is
determined by a 1-cell in $Del({\mathcal{C}})$, i.e. an object $B$ of
${\mathcal{C}}$, and a (co-)multiplication and (co-)unit on that 1-cell, which
are subject to (co-)associativity and (co-)unitality due to ${\mathcal{F}}$
being lax and colax. On the other hand, $c$ of the trivial 2-category is
trivial: it is the identity 2-cell on the identity 1-cell on $*$. Then the
equations (18) recover bialgebra axioms on ${\mathcal{F}}(id_{*})$. (The rest
of axioms of the bilax functor are automatically fulfilled by the braiding,
and do not contribute any additional information.)

If moreover ${\mathcal{F}}$ is a pseudofunctor, then it is trivial: the
obtained bialgebra is isomorphic to the monoidal unit of ${\mathcal{C}}$. This
can be proved considering the lax unitality structure of ${\mathcal{F}}$.

The following result is straightforwardly proved, see also [24, Proposition
3.9].

###### Proposition 4.5

Let ${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ and
${\mathcal{G}}:({\mathcal{K}}^{\prime},d)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime\prime},e)$ be
compatible bilax functors. Then
${\mathcal{G}}{\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime\prime},e)$ is a
compatible bilax functor with a Yang-Baxter operator
$\nu_{g,f}:=\nu^{\mathcal{G}}_{{\mathcal{F}}(g),{\mathcal{F}}(f)}$, where
$\nu^{\mathcal{G}}$ is a Yang-Baxter operator of ${\mathcal{G}}$ and $g,f$ are
1-endocells of ${\mathcal{K}}$.

###  4.2 Bimonads

The theory of monads and comonads in the context of 2-categories was
introduced by Street in [30]. Recall that a monad in ${\mathcal{K}}$ , is a
1-endo-cell $t\in{\mathcal{K}}(A,A)$ endowed with 2-cells
$\mu:tt\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}t$ and
$\eta:id_{A}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}t$ subject to associativity and
unitality conditions. Dually, that is, swapping the direction of the structure
2-cells, one obtains the notion of a comonad in ${\mathcal{K}}$: a 1-endo-cell
endowed with a coassociative and counital comultiplication. In order to
differentiate notations for (co)lax functors and (co)monad structures, we will
represent the multiplication and unit of a monad as well as the
comultiplication and counit of a comonad by:

$\begin{minipage}{24.0pt}
\begin{picture}(24.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(12.0,36.0){\oval(12.0,18.0)[b]} \put(12.0,27.0){\line(0,-2){3.0}}
\put(12.0,27.0){\circle*{4.0}}
\qbezier(12.0,24.0)(12.0,20.0)(12.0,18.0)\qbezier(12.0,18.0)(12.0,16.0)(12.0,12.0)
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\hskip 56.9055pt\begin{minipage}{12.0pt}
\begin{picture}(12.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,24.0){\line(0,1){6.0}} \put(6.0,30.0){\circle*{4.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(0.0,12.0){\line(1,0){0.0}}
\end{picture} {\vskip 9.68745pt} \end{minipage},\hskip
56.9055pt\begin{minipage}{24.0pt}
\begin{picture}(24.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\qbezier(12.0,36.0)(12.0,32.0)(12.0,30.0)\qbezier(12.0,30.0)(12.0,28.0)(12.0,24.0)
\put(12.0,12.0){\oval(12.0,18.0)[t]} \put(12.0,24.0){\circle*{4.0}}
\put(12.0,24.0){\line(0,-1){3.0}} \put(0.0,12.0){\line(1,0){0.0}}
\end{picture} {\vskip 9.68745pt} \end{minipage},\hskip
56.9055pt\begin{minipage}{12.0pt}
\begin{picture}(12.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(6.0,24.0){\line(0,-1){6.0}}
\put(6.0,18.0){\circle*{4.0}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture}
{\vskip 9.68745pt} \end{minipage}.$

One shows in a straightforward manner that lax functors preserve monads and
colax functors preserve comonads. Specifically, for a monad $t$ in
${\mathcal{K}}$ and a lax functor
${\mathcal{F}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$, and a comonad
$d$ in ${\mathcal{K}}$ and a colax functor ${\mathcal{G}}$ we have the
following monad and comonad structures:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(30.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(18.0,36.0){\oval(24.0,18.0)[b]} \put(18.0,27.0){\circle*{4.0}}
\put(18.0,27.0){\line(0,-1){3.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(18.0,48.0){\oval(24.0,18.0)[b]} \put(18.0,39.0){\line(0,-1){3.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\nabla)$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(12.0,25.0){\line(1,0){12.0}} \put(12.0,35.0){\line(1,0){12.0}}
\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\hskip 17.07182pt\begin{minipage}{12.0pt}
\begin{picture}(12.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,24.0){\line(0,1){6.0}} \put(6.0,30.0){\circle*{4.0}}
\put(6.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(18.0,24.0){\line(0,1){6.0}} \put(18.0,30.0){\circle{5.0}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,13.0){\line(0,1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\eta)$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(12.0,13.0){\line(1,0){12.0}} \put(12.0,23.0){\line(1,0){12.0}}
\put(35.0,23.0){\line(-1,0){12.0}} \put(35.0,13.0){\line(-1,0){12.0}}
\put(35.0,13.0){\line(0,1){10.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\hskip 14.22636pt\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(18.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(18.0,36.0){\line(0,-1){12.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\circle*{4.0}} \put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(18.0,48.0){\line(0,-1){12.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\Delta)$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(12.0,25.0){\line(1,0){12.0}} \put(12.0,35.0){\line(1,0){12.0}}
\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\hskip 14.22636pt\begin{minipage}{12.0pt}
\begin{picture}(12.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(6.0,12.0){\line(0,-1){6.0}}
\put(6.0,6.0){\circle*{4.0}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture}
{\vskip 9.68745pt} \end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(18.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,13.0){\line(0,1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\varepsilon)$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(12.0,13.0){\line(1,0){12.0}} \put(12.0,23.0){\line(1,0){12.0}}
\put(35.0,23.0){\line(-1,0){12.0}} \put(35.0,13.0){\line(-1,0){12.0}}
\put(35.0,13.0){\line(0,1){10.0}} \put(18.0,12.0){\line(0,-1){6.0}}
\put(18.0,6.0){\circle{5.0}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture}
{\vskip 9.68745pt} \end{minipage}.$ (21)

We are going to introduce bimonads in 2-categories with respect to Yang-Baxter
operators. Observe that their 1-categorical analogue is different than the
bimonads of [26] and [27] in ordinary categories, but they are a particular
instance of $\tau$-bimonads and bimonads in 2-categories from [11]. Whereas
bimonads in [27] are opmonoidal monads on monoidal categories, bimonads in
[26] are monads and comonads on a not necessarily monoidal category with
compatibility conditions that involve a distributive law $\lambda$. The
bimonads in [11, Definition 7.1] generalize the latter to 2-categories and are
equipped with an analogous 2-cell i.e. distributive law $\lambda$. The