\ar@/_/[rrru]_{\gamma_{gf}'} \ar@{}[u]|-{\Downarrow \Pi_{g,f}'}
}$$

$$\xymatrixrowsep{0.3in}
\xymatrixcolsep{0.3in}
\xymatrix{
{{\mathcal H}}{\overline{g}}\ {\overline{f}} \ar[d]_{\theta{1_{\overline{g}}}{1_{\overline{f}}}} \ar@/_/[rrrd]_{1_{{{\mathcal H}}}\chi_{g,f}}\ar[rr]_{\gamma_{g}{1_{\overline{f}}}} && {\widetilde{g}}{{\mathcal H}}{\overline{f}} \ar[rr]^{{1_{\widetilde{g}}}\gamma_f}&& {\widetilde{g}}{\widetilde{f}}{{\mathcal H}}' \ar[rr]_{{\widetilde{\chi_{g,f}}}1_{{{\mathcal H}}}}&& {\widetilde{gf}}{{\mathcal H}}\ar[d]^{{1_{\widetilde{gf}}}\theta} \\ {{\mathcal H}}'{\overline{g}}\ {\overline{f}} \ar@{}[rrr]<-3mm>|-{\Downarrow c_{\theta,\chi_{g,f}}^{-1}} \ar@/_/[rrrd]_{1_{{{\mathcal H}}'}\chi_{g,f}}
&&&{{\mathcal H}}{\overline{gf}} \ar[d]_{\theta{1_{\overline{gf}}}}\ar@/_/[rrru]_{\gamma_{gf}} \ar@{}[u]|-{\Downarrow \Pi_{g,f}}&&&{\widetilde{gf}}{{\mathcal H}}' \ar@{}[lll]<3mm>|-{\Downarrow \theta_{gf}} \\
&&&{{\mathcal H}}'{\overline{gf}} \ar@/_/[rrru]_{\gamma_{gf}'}
}$$ holds in [**2Cat**]{}$({{\mathcal B}},{{\mathcal B}})$.

Assume that $(\theta, \{\theta_g\}_{g\in G}), (\sigma, \{\sigma_g\}_{g\in G}):({{\mathcal H}}, \gamma, \Pi)\to 
(\widetilde{{\mathcal H}}, \widetilde\gamma, \widetilde\Pi)$ are $G$-pseudonatural transformations. A *$G$-modification* $\alpha: (\theta, \{\theta_g\}_{g\in G})\Rightarrow(\sigma, \{\sigma_g\}_{g\in G})$ is a modification $\alpha: \theta \Rightarrow \sigma$ such that

$$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
{{\mathcal H}}{\overline{g}} \ar[rr]^{\gamma_g } \ar@{}[d]|-{\overset{\alpha \otimes {1_{\overline{g}}}}{\Leftarrow }} \ar@/_3pc/[d]_{\sigma \otimes {1_{\overline{g}}} } \ar@/^3pc/[d]^{\theta \otimes {1_{\overline{g}}} }  && {\widetilde{g}}{{\mathcal H}}\ar[d]^{{1_{\widetilde{g}}}\theta_g} \\
{{\mathcal H}}'{\overline{g}}\  \ar[rr]_{\gamma_{g}'} && {\widetilde{g}}{{\mathcal H}}' \ar@{}[ul]|-{\Downarrow \theta_g}
}$$

$$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
{{\mathcal H}}{\overline{g}} \ar[rr]^{\gamma_g } \ar[d]_{\sigma_g\otimes {1_{\overline{g}}}} \ar@{}[rrd]<-1mm>_{\Downarrow \sigma_g}  && {\widetilde{g}}{{\mathcal H}}\ar@{}[d]|-{\overset{{1_{\widetilde{g}}}\otimes \alpha}{\Leftarrow }} \ar@/_3pc/[d]_{ {1_{\widetilde{g}}}\otimes \sigma } \ar@/^3pc/[d]^{{1_{\widetilde{g}}}\otimes \theta } \\
{{\mathcal H}}'{\overline{g}}\  \ar[rr]_{\gamma_{g}'} && {\widetilde{g}}{{\mathcal H}}' 
}$$

Assume that $({{\mathcal H}}^1, \gamma^1, \Pi^1), ({{\mathcal H}}^2, \gamma^2, \Pi^2), ({{\mathcal H}}^3, \gamma^3, \Pi^3)$ are $G$-pseudofunctors, and $(\theta, \{\theta_g\}_{g\in G}): ({{\mathcal H}}^1, \gamma^1, \Pi^1)\to ({{\mathcal H}}^2, \gamma^2, \Pi^2)$, $(\sigma, \{\sigma_g\}_{g\in G}): ({{\mathcal H}}^2, \gamma^2, \Pi^2)\to ({{\mathcal H}}^3, \gamma^3, \Pi^3)$ are $G$-pseudonatural transformations. The composition $$(\sigma, \{\sigma_g\}_{g\in G}) \circ  (\theta, \{\theta_g\}_{g\in G})=(\rho, \{\rho_g\}_{g\in G})$$ 