ScalingStacks

Let $\Phi:{\mathcal{W}}\to{\mathcal{W}}^{\prime}$ be a differential functor and $\varphi_{i}:\Phi E_{i}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}_{i}\Phi$ be closed isomorphisms of functors making $(\Phi,\varphi_{i})$ into morphisms of $2$-representations for $i\in\{1,2\}$. Assume

(4.3.4)

$$(E^{\prime}_{1}\varphi_{2})\circ(\varphi_{1}E_{2})\circ(\Phi\sigma)=(\sigma^{\prime}\Phi)\circ(E^{\prime}_{2}\varphi_{1})\circ(\varphi_{2}E_{1}):\Phi E_{2}E_{1}\to E^{\prime}_{1}E^{\prime}_{2}\Phi.$$