ScalingStacks

Proof. The fact that $\tau$ defines an endomorphism of $(A,A)$-bimodules of $E^{2}$ satisfying the appropriate relations follows from the fact that it agrees with the endomorphism of ${\mathcal{E}}^{2}$ defining the $2$-representation on $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$. We deduce that $(E,\tau)$ is a $2$-representation on $A$ and $\Phi$ is a morphism of $2$-representations.

Note that $E$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module if $E_{1}$ and $E_{2}$ are finitely generated and projective $B^{\operatorname{opp}\nolimits}$-modules. ∎