&  &  & {\widetilde{f}}{\widetilde{g}}{\widetilde{h}}{{\mathcal H}}\ar[rd]^{{\widetilde{\chi_{f,g}}}1_{{\widetilde{h}}{{\mathcal H}}}} {\ar @{} [dl] |{c_{\widetilde\chi_{f,g},\gamma_h}}}&  \\ 
 {{\mathcal H}}{\overline{f}}{\overline{g}}{\overline{h}} \ar[rr]^{1_{{{\mathcal H}}}\chi_{f,g}1_{{\overline{h}}}} \ar[ru]^{\gamma_f 1_{{\overline{g}}{\overline{h}}}}\ar[rd]_{1_{{{\mathcal H}}{\overline{f}}}\chi_{g,h}} &  & {{\mathcal H}}{\overline{fg}}\ {\overline{h}} \ar[rr]^{\gamma_{fg}1_{{\overline{h}}}} \ar[ddr]^{1_{{{\mathcal H}}}\chi_{fg,h}} &  &  {\widetilde{fg}}{{\mathcal H}}{\overline{h}} {\ar @{} [dd]+<6mm> |{\Downarrow\Pi_{fg,h}}} \ar[rr]^{1_{{\widetilde{fg}}}\gamma_h}&  & {\widetilde{fg}}{\widetilde{h}}{{\mathcal H}}\ar[ld]^{{\widetilde{\chi_{fg,h}}}1_{{{\mathcal H}}}}\\ 
  & {{\mathcal H}}\overline{f}\ \overline{gh} {\ar @{} [ur] |{\overset{1_{{{\mathcal H}}}{{\otimes}}\omega_{f,g,h}}{\Leftarrow}}} \ar[rrd]_{1_{{{\mathcal H}}}\chi_{f,gh}} &  &  &  & {\widetilde{fgh}}{{\mathcal H}}&  \\ 
  &  &  & {{\mathcal H}}\overline{fgh} \ar[rru]_{\gamma_{fgh}} &  &  & 
}$$ holds in [**2Cat**]{}$({{\mathcal B}},{{\mathcal B}})$. In the above diagrams, we are using the comparison constraints $c$ defined in .

Assume that $({{\mathcal H}},\gamma, \Pi), ({{\mathcal H}}',\gamma', \Pi')$ are $G$-pseudofunctors. A $G$-*pseudonatural* transformation is a pair $(\theta, \{\theta_g\}_{g\in G})$, where $\theta: {{\mathcal H}}\to {{\mathcal H}}'$ is a pseudonatural transformation, and $\theta_g$ are invertible modifications $$\xymatrixrowsep{0.6in}
\xymatrixcolsep{0.6in}
\xymatrix{
{{\mathcal H}}{\overline{g}} \ar[r]^{\gamma_g } \ar[d]_{\theta{{\otimes}}{1_{\overline{g}}} } & {\widetilde{g}}{{\mathcal H}}\ar[d]^{{1_{\widetilde{g}}}{{\otimes}}\theta} \\
{{\mathcal H}}'{\overline{g}}\  \ar[r]^{\gamma_{g}'} & {\widetilde{g}}{{\mathcal H}}' \ar@{}[ul]|-{\Downarrow \theta_g}
}$$

such that for all $g,f \in G$, the equation $$\xymatrixrowsep{0.3in}
\xymatrixcolsep{0.3in}
\xymatrix{
{{\mathcal H}}{\overline{g}}\ {\overline{f}} \ar@{}[rrd]|-{\Downarrow \theta_g{1_{\overline{f}}}} \ar[rr]^{\gamma_g {1_{\overline{f}}}} \ar[d]_{\theta {1_{\overline{g}}} {1_{\overline{f}}}} &&  {\widetilde{g}}{{\mathcal H}}{\overline{f}} \ar@{}[rrd]|-{\Downarrow {1_{\widetilde{g}}}\theta_f} \ar[d]_{{1_{\widetilde{g}}}\theta{1_{\overline{f}}}} \ar[rr]^{{1_{\widetilde{g}}}\gamma_f} && {\widetilde{g}}{\widetilde{f}}{{\mathcal H}}\ar@{}[rrd]|-{\Downarrow c_{\theta,{\widetilde{\chi_{g,f}}}}} \ar[d]_{{1_{\widetilde{g}}}{1_{\widetilde{f}}}\theta}  \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@/_/[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}}' & \\
&&&{{\mathcal H}}'{\overline{gf}} 