between ${{\mathcal B}}$ and $\widetilde{{{\mathcal B}}}$ is a triple $({{\mathcal H}}, \gamma, \Pi)$, where

-   ${{\mathcal H}}:{{\mathcal B}}\to \widetilde{{{\mathcal B}}}$ is a unital pseudofunctor,

-   for any $g\in G$, pseudonatural equivalences $\gamma_g:{{\mathcal H}}\circ F_g \to \widetilde F_g\circ {{\mathcal H}},$

-   invertible modifications $$\xymatrix{
    &{\widetilde{f}}{{\mathcal H}}{\overline{g}} \ar[rr]^{1_{{\widetilde{f}}}\otimes \gamma_g}&& {\widetilde{f}}{\widetilde{g}}{{\mathcal H}}\ar[rd]^{{\widetilde{\chi_{f,g}}}\otimes 1_{{{\mathcal H}}}}\\
    {{\mathcal H}}{\overline{f}}\ {\overline{g}} \ar[ru]^{\gamma_f\otimes 1_{{\overline{g}}}} \ar[rrd]_{1_{{{\mathcal H}}}\otimes \chi_{f,g}}&&&& {\widetilde{fg}}{{\mathcal H}}\\
    &&{{\mathcal H}}{\overline{fg}} \ar[rru]_{\gamma_{fg}} {\ar @{} [uu] |{\Downarrow\Pi_{f,g}}}&&
    }$$

such that such that for all $f,g,h\in G$ $$\begin{aligned}
\label{G-funct1} \gamma_1={\mbox{\rm id\,}}_{{\mathcal H}}, \quad \Pi_{g,1}={\mbox{\rm id\,}}_{\gamma_g}=\Pi_{1,g},\end{aligned}$$ $$\label{Pi-axiom}
\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  &  & {\widetilde{f}}{\widetilde{g}}{{\mathcal H}}{\overline{h}} \ar[rrd]^{1_{{\widetilde{f}}}1_{{\widetilde{g}}}\gamma_h } {\ar @{} [dd] |{\Downarrow1_{{\widetilde{f}}}{{\otimes}}\Pi_{g,h}}} &  &  &  \\ 
  & {\widetilde{f}}{{\mathcal H}}{\overline{g}}{\overline{h}}  \ar[rru]^{1_{{\widetilde{f}}}\gamma_g1_{{\overline{h}}}}\ar[rd]^{1_{{\widetilde{f}}{{\mathcal H}}}\chi_{g,h}} &  &  &  & {\widetilde{f}}{\widetilde{g}}{\widetilde{h}}{{\mathcal H}}\ar[ld]_{1_{{\widetilde{f}}}{\widetilde{\chi_{g,h}}}1_{{{\mathcal H}}}}\ar[rd]^{{\widetilde{\chi_{f,g}}}1_{{\widetilde{h}}{{\mathcal H}}}} &  \\ 
 {{\mathcal H}}{\overline{f}}{\overline{g}}{\overline{h}} {\ar @{} [rr] |{c_{\gamma_f,\chi_{g,h}}}} \ar[ru]^{\gamma_f 1_{{\overline{g}}\ {\overline{h}}}}\ar[rd]_{1_{{{\mathcal H}}{\overline{f}}}\chi_{g,h}} &  & {\widetilde{f}}{{\mathcal H}}{\overline{gh}}\ar[rr]^{1_{{\widetilde{f}}}\gamma_{gh}} &  & {\widetilde{f}}{\widetilde{gh}}{{\mathcal H}}\ar[rd]_{{\widetilde{\chi_{f,gh} 1_{{{\mathcal H}}}}}} &  & {\widetilde{fg}}{\widetilde{h}}{{\mathcal H}}\ar[ld]^{{\widetilde{\chi_{fg,h}}}1_{{{\mathcal H}}}}\\ 
  & {{\mathcal H}}\overline{f}\ \overline{gh} \ar[ru]_{\gamma_f 1_{{\overline{gh}}}}\ar[rrd]_{\chi_{f,gh}} &  &  &  & {\widetilde{fgh}}{{\mathcal H}}{\ar @{} [uu] |{\overset{\widetilde\omega_{f,g,h}}{\Leftarrow}}} &  \\ 
  &  &  & {{\mathcal H}}\overline{fgh} \ar[rru]_{\gamma_{fgh}} {\ar @{} [uu] |{\Downarrow\Pi_{f,gh}}}&  &  & 
}$$

$$\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  &  & {\widetilde{f}}{\widetilde{g}}{{\mathcal H}}{\overline{h}} \ar[ddr]^{{\widetilde{\chi_{f,g}}}1_{{{\mathcal H}}{\overline{h}}}} \ar[rrd]^{1_{{\widetilde{f}}}1_{{\widetilde{g}}}\gamma_h } &  &  &  \\ 
  & {\widetilde{f}}{{\mathcal H}}{\overline{g}}{\overline{h}} {\ar @{} [rrr] |{\Downarrow \Pi_{f,g}{{\otimes}}1_{{\overline{h}}}}} \ar[rru]^{1_{{\widetilde{f}}}\gamma_g1_{{\overline{h}}}} &  