The Stacks project

Proposition 59.47.1. Let $f : X \to Y$ be a morphism of schemes which is integral and universally injective.

1. The functor

   $$f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$$

   is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $Y_{\acute{e}tale}$ whose restriction to $Y \setminus f(X)$ is isomorphic to $*$, and

2. the functor

   $$f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \longrightarrow \textit{Ab}(Y_{\acute{e}tale})$$

   is fully faithful and its essential image is those abelian sheaves on $Y_{\acute{e}tale}$ whose support is contained in $f(X)$.

In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$.