&  &  &  & {\overline{g}}\ {\overline{hfk}}\ar[rruu]_{\chi_{g,hfk}} {\ar @{} [lluu] |{\Downarrow\omega_{g,h,fk}}} &  & 
}$$

We say that a group $G$ acts *trivially* on ${{\mathcal B}}$ if the weak monoidal homomorphism $({{\mathcal F}},\chi,\omega,\iota,\kappa,\zeta): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$ is the trivial one. This means that for any $g,h\in G$, the pseudofunctors $F_g$ are the identity, $\chi_{g,h}$ are the identity pseudonatural transformations and all the modifications are identities.

\[unital-act\] An action $({{\mathcal F}},\chi,\omega,\iota,\kappa,\zeta): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$ is called *unital* if $F_g$ is a unital pseudofunctor, $F_1={\mbox{\rm Id\,}}_{{\mathcal B}}$, and $\chi_{g,1}={\mbox{\rm id\,}}_{F_g}=\chi_{1,g}$, $\kappa_g={\mbox{\rm id\,}}=\zeta_g$ for any $g\in G$. A unital $G$-action will be denoted simply by $({{\mathcal F}},\chi,\omega)$.

\[strict-act\] An action $({{\mathcal F}},\chi,\omega,\iota,\kappa,\zeta): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}})$ is called *strict* if each pseudofunctor $F_g$ is a 2-functor, and $F_g\circ F_h=F_{gh}$, and the pseudonatural transformations $\chi_{g,h}$ and the modifications $\omega_{g,h,f}$ are the identities for any $g,h,f\in G$.

A similar argument as in [@Ga2 Proposition 3.1] applied in this case, allows us to consider only unital actions. Assume that ${{\mathcal B}}, {{\mathcal B}}'$ are 2-categories equipped with unital actions of a group $G$ via $$({{\mathcal F}},\chi,\omega): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}({{\mathcal B}}, {{\mathcal B}}),\,\, (\widetilde {{\mathcal F}},\widetilde \chi,\widetilde \omega): {{\underline{\underline{G}}}}\to {\textbf{2Cat}}(\widetilde{{{\mathcal B}}}, \widetilde{{{\mathcal B}}}).$$

A *$G$-pseudofunctor* 