Lemma 59.104.4. Let $f : X \to Y$ be a morphism of schemes. Let $Z \to Y$ be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors

and

are equivalences of categories, then

is an equivalence.

Lemma 59.104.4. Let $f : X \to Y$ be a morphism of schemes. Let $Z \to Y$ be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors

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

and

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

are equivalences of categories, then

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

is an equivalence.

**Proof.**
Formal consequence of the definitions and Lemmas 59.104.2 and 59.104.3. Details omitted.
$\square$

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