ScalingStacks

Proof. The vanishing of $d(v)_{11}$ and $d(v)_{22}$ follows from $d(\tau_{1})=\operatorname{id}\nolimits$ and $d(\tau_{2})=\operatorname{id}\nolimits$. The vanishing of $d(v)_{12}$ is clear. Finally, the vanishing of $d(v)_{21}$ follows from the commutativity of the diagram (5.3.3). Since $d(v)=0$, we have obtained a structure of differential $(T_{B}(E_{1}^{\vee}E_{2}),A)$-bimodule on $E$.

The object of $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$ corresponding to $A$ via Lemma 5.3.2 is $(A,u)$. We have ${\mathcal{E}}(A,u)=(E,v)$, where ${\mathcal{E}}$ is the endofunctor defining the $2$-representation on $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$. Since $(E,v)$ is an object of $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$, it follows that the action of $T_{B}(E_{1}^{\vee}E_{2})$ on $E$ factors through an action of $A$. So, $E$ has a structure of differential $(A,A)$-bimodule and we have an isomorphism of functors $\Phi{\mathcal{E}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(E\otimes_{A}-)\Phi:\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\to A\operatorname{\!-diff}\nolimits$. ∎