Lemma 6.32.4. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure with final object $0$. The functor

$i_* : \mathop{\mathit{Sh}}\nolimits (Z, \mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})$

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

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