See for example [@GPS].

Explicitly, an action of $G$ on a 2-category ${{\mathcal B}}$ consists of the following data:

-   A family of pseudofunctors $F_g:{{\mathcal B}}\to {{\mathcal B}}$, $g\in G$,

-   pseudonatural equivalences $(\chi_{g,h}, \chi^0_{g,h}): F_g\circ F_h\to F_{gh}$, $g, h\in G$,

-   a pseudonatural equivalence $\iota: {\mbox{\rm Id\,}}_{{\mathcal B}}\to F_1$,

-   for any $g,h,f\in G$ invertible modifications $$\omega_{g,h,f}: \chi_{gh,f} \circ (\chi_{g,h} {{\otimes}}{\mbox{\rm id\,}}_{F_f }) \Rightarrow \chi_{g,hf}\circ ({\mbox{\rm id\,}}_{F_g} {{\otimes}}\chi_{h,f}),$$ $$\kappa_g: \chi_{1,g} \circ (\iota{{\otimes}}{\mbox{\rm id\,}}_{F_g} ) \Rightarrow {\mbox{\rm id\,}}_{F_g}, \quad \zeta_g: \chi_{g,1}\circ ({\mbox{\rm id\,}}_{F_g}{{\otimes}}\iota)\Rightarrow
     {\mbox{\rm id\,}}_{F_g},$$

such that for any 0-cell $A$ $$\begin{aligned}
\label{mn1}
1_{(\chi^0_{g,f})_A} \circ F_g(\kappa_f)_A (\omega_{g,1,f})_A= 1_{(\chi^0_{g,f})_A} \circ ( \zeta_g)_{F_f(A)},\end{aligned}$$ $$\begin{aligned}
\label{mn2}\begin{split} &\big( {\mbox{\rm id\,}}_3\circ (F_g(\omega_{h,f,k})_A) \big)\big(\omega_{g,hf,k}\circ {\mbox{\rm id\,}}_2 \big)
\big({\mbox{\rm id\,}}_{(\chi^0_{ghf,k})_A}\circ (\omega_{g,h,f})_{F_k(A)}\big)=\\
&=\big((\omega_{g,h,fk})_A\circ  {\mbox{\rm id\,}}_4)\big)
\big({\mbox{\rm id\,}}_5\circ (\chi_{g,h})_{ \chi^0_{f,k}}\big)\big((\omega_{gh,f,k})_A\circ {\mbox{\rm id\,}}_6\big),
\end{split}\end{aligned}$$ for any $g,h,f,k\in G$. Where, $${\mbox{\rm id\,}}_{2}=1_{F_g( \chi^0_{h,f})_{F_k(A)}},\quad  {\mbox{\rm id\,}}_{3}=1_{(\chi^0_{g,hfk})_A}, \quad {\mbox{\rm id\,}}_4=1_{F_g F_f( \chi^0_{h,k})_A},$$ $${\mbox{\rm id\,}}_{5}=1_{(\chi^0_{gh,fk})_A}, \quad  {\mbox{\rm id\,}}_{6}= 1_{(\chi^0_{g,h})_{F_fF_k(A)}}.$$ In equation , we are omitting the associativity isomorphisms of the pseudofunctors $F_g$. In the following diagrams we shall denote by ${\overline{g}}$ the pseudofunctor $F_g$, the composition of functors as juxtaposition and the tensor product of pseudonatural transformations also by juxtaposition. Diagrammatically, we have modifications $$\xymatrixrowsep{0.5in}
\xymatrixcolsep{0.5in}
\xymatrix{
{\overline{g}}\ {\overline{h}}\ {\overline{f}} \ar[d]_{1_{{\overline{g}}}\otimes \chi_{h,f}} \ar[rr]^{\chi_{g,h}\otimes 1_{{\overline{f}}}}&& 