any 0-cells $A, B\in {{\mathcal B}}$, and $X\in {{\mathcal B}}(A,B)$ $$\label{tensor-prod-relativ} (V{{\otimes}}W)_A=V_A\circ W_A,\quad
(\sigma{{\otimes}}\tau)_X = (\sigma_X\circ {\mbox{\rm id\,}}_{W_A})({\mbox{\rm id\,}}_{V_B}\circ \tau_X).$$ If $(V,\sigma), (V',\sigma'), (W,\tau), (W',\tau')\in {{\mathcal Z}}({{\mathcal H}})$ are objects, and $f:(V,\sigma)\to (V',\sigma'),$ $f':(W,\tau), (W',\tau')$ are morphisms in ${{\mathcal Z}}({{\mathcal H}})$, then $f{{\otimes}}f':(V,\sigma){{\otimes}}(V',\sigma') \to (W,\tau){{\otimes}}(W',\tau')$ is defined by $$(f{{\otimes}}f')_A=f_A\circ f'_A,$$ for any 0-cell $A$. The unit $({ \mathbf{1}},\iota)\in{{\mathcal Z}}({{\mathcal H}})$ is the object $${ \mathbf{1}}_A=I_A, \quad \iota_X={\mbox{\rm id\,}}_X,$$ for any 0-cells $A, B$ and any 1-cell $X\in {{\mathcal B}}(A,B).$ The center ${{\mathcal Z}}({\mbox{\rm Id\,}}_{{\mathcal B}})$ of the identity pseudofunctor ${\mbox{\rm Id\,}}_{{\mathcal B}}:{{\mathcal B}}\to{{\mathcal B}}$ is denoted as ${{\mathcal Z}}({{\mathcal B}})$, and it coincides with the definition presented in [@MS].

Group actions on 2-categories {#Section:group-2-cat}
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Assume $G$ is a group and ${{\mathcal B}}$ is a 2-category. We shall denote by ${{\underline{\underline{G}}}}$ the 2-category that has 0-cells the elements of the group $G$. For any pair $g,h\in G$ $${{\underline{\underline{G}}}}(g,h)=\begin{cases} \text{ the unit category, if}\quad g=h\\
\emptyset\,  \text{ if }\quad g\neq h.
  \end{cases}$$ Moreover, ${{\underline{\underline{G}}}}$ is a monoidal 2-category, see [@GPS]. Since ${\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$ is also a monoidal 2-category, we define an *action* of $G$ on ${{\mathcal B}}$ as a weak monoidal homomorphism $({{\mathcal F}},\chi,\omega,\iota,\kappa,\zeta): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$. 