The Stacks project

Proof. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $U$ be an object of $X_{\acute{e}tale}$. By assumption we can find a covering $\{ W_ i \to U\} $ in $X_{\acute{e}tale}$ such that each $W_ i$ is an open and closed subscheme of $X \times _ Y V_ i$ for some object $V_ i$ of $Y_{\acute{e}tale}$. The sheaf condition shows that

and that $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i) = f_{small, *}\mathcal{F}(V_ i)$. Hence it is clear that $f_{small, *}$ reflects injections.

Next, suppose that $a : \mathcal{G} \to \mathcal{F}$ is a map of abelian sheaves such that $f_{small, *}a$ is surjective. Let $s \in \mathcal{F}(U)$ with $U$ as above. With $W_ i$, $V_ i$ as above we see that it suffices to show that $s|_{W_ i}$ is étale locally the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i)$ it suffices to show that for any $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$ any element $s \in \mathcal{F}(X \times _ Y V)$ is étale locally on $X \times _ Y V$ the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(X \times _ Y V) = f_{small, *}\mathcal{F}(V)$ we see by assumption that there exists a covering $\{ V_ j \to V\} $ such that $s$ is the image of $s_ j \in f_{small, *}\mathcal{G}(V_ j) = \mathcal{G}(X \times _ Y V_ j)$. This proves $f_{small, *}$ reflects surjections.

Parts (2), (3) follow formally from part (1), see Modules on Sites, Lemma 18.15.1. $\square$