Lemma 59.104.3. Let $f : X \to Y$ be a surjective proper morphism of schemes. The functor

is an equivalence of categories.

Lemma 59.104.3. Let $f : X \to Y$ be a surjective proper morphism of schemes. The functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

**Proof.**
The exact same proof as given in Lemma 59.104.2 works, except the appeal to Lemma 59.55.4 should be replaced by an appeal to Lemma 59.91.5.
$\square$

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