The Stacks project

Lemma 7.11.2. The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. A map of sheaves is an isomorphism if and only if it is both injective and surjective.

Proof. Omitted. $\square$

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