Lemma 65.18.3. The functor

is a special cocontinuous functor (Sites, Definition 7.29.2) and hence induces an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$.

Lemma 65.18.3. The functor

\[ X_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}}, \quad U/X \longmapsto U/X \]

is a special cocontinuous functor (Sites, Definition 7.29.2) and hence induces an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$.

**Proof.**
We have to show that the functor satisfies the assumptions (1) – (5) of Sites, Lemma 7.29.1. It is clear that the functor is continuous and cocontinuous, which proves assumptions (1) and (2). Assumptions (3) and (4) hold simply because the functor is fully faithful. Assumption (5) holds, because an algebraic space by definition has a covering by a scheme.
$\square$

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