The Stacks project

Lemma 59.44.5. Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is universally injective and integral (for example a closed immersion). Then

1. $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ reflects injections and surjections,

2. $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ commutes with pushouts and coequalizers (and more generally finite connected colimits),

3. $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves),

4. the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any sheaf (of sets or of abelian groups) $\mathcal{F}$ on $X_{\acute{e}tale}$,

5. the functor $f_{small, *}$ is faithful (on sheaves of sets and on abelian sheaves),

6. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact, and

7. the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ is almost cocontinuous.