# The Stacks Project

## Tag: 06WQ

This tag has label stacks-sheaves-lemma-stackification and it points to

The corresponding content:

Lemma 64.12.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then the morphism of topoi $f : \mathop{\textit{Sh}}\nolimits(\mathcal{X}_{fppf}) \to \mathop{\textit{Sh}}\nolimits(\mathcal{Y}_{fppf})$ 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 7.26.2 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 7.26.1 hold. $\square$

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

\begin{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 \ref{sites-definition-special-cocontinuous-functor}
which will prove the lemma. By
Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial}
the functor $f$ is continuous and cocontinuous. By
Stacks, Lemma \ref{stacks-lemma-stackify}
we see that conditions (3), (4), and (5) of
Sites, Lemma \ref{sites-lemma-equivalence}
hold.
\end{proof}


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