The Stacks project

Lemma 67.53.7. Let $S$ be a scheme. Let $f : Y \to X$ be a universally injective, integral morphism of algebraic spaces over $S$.

1. The functor

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

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

2. the functor

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

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

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