is defined as follows. The pseudonatural transformation $\rho=\sigma\circ \theta$. For any 0-cell $A\in {{\mathcal B}}$ and any $g\in G$ $$(\rho_g)_A= \big((\sigma_g)_A\circ {\mbox{\rm id\,}}_{  \theta^0_{F_g(A)}} \big)
\big( {\mbox{\rm id\,}}_{\widetilde F_g(\sigma^0_A)}\circ (\theta_g)_A)\big).$$ Here, we are also ommiting the associativity constraints of the pseudofunctor $F_g$. The composition of modifications of $G$-categories is the usual composition of modifications.

${\textbf{2Cat}}^G({{\mathcal B}}, \widetilde{{\mathcal B}})$ is the 2-category in which 0-cells are pseudofunctors of $G$-categories, 1-cells are pseudonatural transformations of $G$-categories and 2-cells are modifications of $G$-categories.

The next result is a consequence of [@GPS Corollary 8.3].

${\textbf{2Cat}}^G({{\mathcal B}}, \widetilde{{\mathcal B}})$ is a 2-category.

We say that the 2-categories ${{\mathcal B}}$ and $ \widetilde{{\mathcal B}}$ are $G$-*biequivalent* if there exists a $G$-pseudofunctor ${{\mathcal H}}:{{\mathcal B}}\to \widetilde{{\mathcal B}}$ that is also a biequivalence.

\[transport\] Let ${{\mathcal B}}$ be a 2-category with an action of $G$ given by $({{\mathcal F}},\chi,\omega)$. Let ${{\mathcal H}}:{{\mathcal B}}\to {{\mathcal B}}'$ be a biequivalence, $$L_g:{{\mathcal B}}'\to {{\mathcal B}}', \  \  \gamma_g:{{\mathcal H}}\circ F_g\to L_g\circ {{\mathcal H}}$$ a $G$-indexed family of pseudofunctors and pseudonatural equivalences, respectively. Then, there is a way to endowed ${{\mathcal B}}'$ with a $G$-action $(L,\chi',\omega')$ such that $({{\mathcal H}},\gamma, \Pi ):{{\mathcal B}}\to {{\mathcal B}}'$ is a $G$-biequivalence .

Since $\gamma_g$ and 