The Stacks project

Lemma 59.44.1. Let $f : X \to Y$ be a morphism of schemes. Assume (C) holds. Then the functor $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ reflects injections and surjections.

Proof. Follows from Sites, Lemma 7.41.4. We omit the verification that property (C) implies that the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ satisfies the assumption of Sites, Lemma 7.41.4. $\square$

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