unital action of $G$, $({{\mathcal F}},\chi,\omega): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$. By Corollary \[action with 2-functors\] we can assume that $F_g$ is a 2-functor for any $g\in G$. We shall first construct a 2-category ${{\mathcal B}}[G]$ with a strict action of $G$.

Objects of ${{\mathcal B}}[G]$ are triples $(A,\theta,\alpha)$, where $A=\{A_g\}_{g}$ is a $G$-indexed family of objects, $\theta=\{\theta_{g,h}:F_g(A_h)\to A_{gh}\}_{g,h\in G}$ is a $G\times G$-indexed family of 1-cell equivalences and $$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
F_{g}F_{h}(A_{f}) \ar[d]_{F_g(\theta_{h,f})} \ar[rr]^{(\chi_{g,h}^0)_{A_f}}&& F_{gh}(A_{f}) \ar[d]^{\theta_{gh,f}}\\
F_g(A_{hf}) \ar[rr]_{\theta_{g,hf}}&& A_{ghf} {\ar @{} [llu] |{\Downarrow\alpha_{g,h,f}}},
}$$ a $G\times G\times G$-index family of isomorphism 2-cells, such $$\theta_{1,g}=I_{A_g}, \quad \alpha_{1,h,f}=\operatorname{id}, \quad \alpha_{g,1,f}=\operatorname{id}$$ that for all $g,h,f,k$, and equation $$\label{obj-axiom} 
\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  & {\overline{gh}}\ {\overline{f}}\ A_k\ar[rrrr]^{\chi_{gh,f}^0} {\ar @{} [ddrr] |{\Downarrow\omega_{g,h,f}}} &  &  &  & {\overline{ghf}}\ A_k\ar[rrdd]^{\theta_{ghf,k}} &  &  \\ 
  &  &  &  &  &  &  &  &  \\ 
 {\overline{g}}\ {\overline{h}}\ {\overline{f}}\ A_{k}\ar[rruu]^{\chi_{g,h}^0\otimes 1_{{\overline{f}}}}  \ar[rrdd]_{1_{{\overline{g}}}\otimes 1_{{\overline{h}}}\otimes \theta_{f,k}}\ar[rrrr]^{1_{{\overline{g}}}\otimes \chi_{h,f}^0} &  &  &  & {\overline{g}}\ {\overline{hf}}\ A_k\ar[rruu]^{\chi_{g,hf}^0}\ar[rrdd]^{1_{{\overline{g}}}\otimes \theta_{hf,k}} &  &  &  & A_{ghfk} {\ar @{} [llll] |{\Downarrow \alpha_{g,hf,k}}}\\ 
  &  &  &  &  &  &  &  &  \\ 
  &  &  {\overline{g}} \ {\overline{h}}\ A_{fk} {\ar @{} [rrruu] |{\Downarrow 1_{{\overline{g}}}\otimes \alpha_{h,f,k}}} \ar[rrrr]_{1_{{\overline{g}}}\otimes \theta_{h,fk}} &  &  &  & {\overline{g}}\ A_{hfk}\ar[rruu]_{\theta_{g,hfk}} &  & 
}$$

$$\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  & 