$(a_{\alpha,\beta,\gamma})_A=\operatorname{id}_{\alpha_{F'H'(A)}}\circ G_2(\beta_{H'(A)},F(\gamma_A))$. It is easy to see that $a$ satisfies the pentagonal identity.

Finite tensor categories {#subsection:tensorcat}
------------------------

A (strict) monoidal category is a 2-category with one single 0-cell. A *finite tensor category over* ${{\Bbbk}}$ is a finite ${{\Bbbk}}$-linear abelian rigid monoidal category ${{\mathcal C}}$ such that the tensor product functor $\otimes : {{\mathcal C}}\times {{\mathcal C}}\to {{\mathcal C}}$ is ${{\Bbbk}}$-linear in each variable. The reader is referred to [@eo].

Suppose ${{\mathcal C}}$ and ${{\mathcal D}}$ are strict tensor categories. A *monoidal functor* $(F, \xi, \phi):{{\mathcal C}}\to 
{{\mathcal D}}$ is a pseudofunctor between the corresponding 2-categories. Explicitly, it consists of a functor $F:{{\mathcal C}}\to {{\mathcal D}}$, natural isomorphisms $\xi_{X,Y}:F(X){{\otimes}}F(Y)\to F(X{{\otimes}}Y),$ $X, Y\in {{\mathcal C}}$, and isomorphism $\phi:{ \mathbf{1}}\to
F({ \mathbf{1}})$, satisfying certain axioms. If $(F,\xi,\phi), (F',\xi',\phi')$ are monoidal functors , a [ *natural monoidal transformation*]{} $\theta:(F,\xi,\phi)\to
(F',\xi',\phi')$ is a natural transformation $\theta:F\to
F'$, such that for any pair of objects $X, Y$ $$\label{tensort1} \theta_{{ \mathbf{1}}}\phi=\phi',\quad
\theta_{X{{\otimes}}Y}\xi_{X,Y}=\xi'_{X,Y} (\theta_{X}{{\otimes}}\theta_Y).$$

The endomorphism category of a pseudofunctor {#relative-center}
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If ${{\mathcal B}}$ is a 2-category, the monoidal category $${{\mathcal Z}}({{\mathcal B}})={\textbf{2Cat}}({{\mathcal B}},{{\mathcal B}})({\mbox{\rm Id\,}}_{{{\mathcal B}}},{\mbox{\rm Id\,}}_{{{\mathcal B}}})$$ is exactly the center of ${{\mathcal B}}$, *i.e.*, the obvious generalization of the center construction of a 