ScalingStacks

Proof of Theorem 8.2.1. Lemma 8.2.12 shows that $q$ factors through an isomorphism $G/\!\sim\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Id}\nolimits_{{\mathcal{S}}_{M}^{\bullet}(Z_{\xi})}$. Since the restriction of $q$ to $C$ is surjective (Lemma 8.2.10), it follows that $q$ induces an isomorphism $C/\!\sim\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Id}\nolimits_{{\mathcal{S}}_{M}^{\bullet}(Z_{\xi})}$.

Recall that $\mu_{i}:(R_{\xi_{2}^{-}}^{\bullet}L^{\bullet}_{\xi_{1}^{+}})^{i}\to G_{i}$ has image $C_{i}$, hence $\mu_{i}$ induces an isomorphism $(R_{\xi_{2}^{-}}^{\bullet}L^{\bullet}_{\xi_{1}^{+}})^{i}/K_{i}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{i}/\!\sim$. As a consequence, the canonical surjective map $T^{*}(R_{\xi_{2}^{-}}^{\bullet}L^{\bullet}_{\xi_{1}^{+}})\to\operatorname{Id}\nolimits_{\Delta_{E}({\mathcal{S}}^{\bullet}_{M}(Z))}$ factors through a surjective map $C/\!\sim\ \to\operatorname{Id}\nolimits_{\Delta_{E}({\mathcal{S}}^{\bullet}_{M}(Z))}$. Since the restriction of $q$ to $C$ factors through $\operatorname{Id}\nolimits_{\Delta_{E}({\mathcal{S}}^{\bullet}_{M}(Z))}$, we deduce that we have an isomorphism $\operatorname{Id}\nolimits_{\Delta_{E}({\mathcal{S}}^{\bullet}_{M}(Z))}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Id}\nolimits_{{\mathcal{S}}_{M}^{\bullet}(Z_{\xi})}$. ∎