Proposition 59.46.4. Let $i : Z \to X$ be a closed immersion of schemes.
The functor
\[ i_{small, *} : \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]
is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and
the functor
\[ i_{small, *} : \textit{Ab}(Z_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]
is fully faithful and its essential image is those abelian sheaves on $X_{\acute{e}tale}$ whose support is contained in $Z$.
In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.
Proof.
Let's discuss the case of sheaves of sets. For any sheaf $\mathcal{G}$ on $Z$ the morphism $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism by Lemma 59.46.3 (and Theorem 59.29.10). This implies formally that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. It is clear that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} \cong *$ where $U = X \setminus Z$. Conversely, suppose that $\mathcal{F}$ is a sheaf of sets on $X$ such that $\mathcal{F}|_{U_{\acute{e}tale}} \cong *$. Consider the adjunction mapping
\[ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F} \]
Combining Lemmas 59.46.3 and 59.36.2 we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical.
$\square$
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