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.
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
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, *}.
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