The Stacks project

Lemma 96.13.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then the morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{fppf})$ is an equivalence.

Proof. We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. We claim that $f : \mathcal{X} \to \mathcal{Y}$ is a special cocontinuous functor, see Sites, Definition 7.29.2 which will prove the lemma. By Stacks, Lemma 8.10.3 the functor $f$ is continuous and cocontinuous. By Stacks, Lemma 8.8.1 we see that conditions (3), (4), and (5) of Sites, Lemma 7.29.1 hold. $\square$

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