Lemma 6.16.3. Let $X$ be a topological space.

Epimorphisms (resp. monomorphisms) in the category of presheaves are exactly the surjective (resp. injective) maps of presheaves.

Epimorphisms (resp. monomorphisms) in the category of sheaves are exactly the surjective (resp. injective) maps of sheaves, and are exactly those maps which are surjective (resp. injective) on all the stalks.

The sheafification of a surjective (resp. injective) morphism of presheaves of sets is surjective (resp. injective).

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