The Stacks project

Coverings become surjective after sheafification.

Lemma 7.12.4. Let $\mathcal{C}$ be a site. If $\{ U_ i \to U\} _{i \in I}$ is a covering of the site $\mathcal{C}$, then the morphism of presheaves of sets

\[ \coprod \nolimits _{i \in I} h_{U_ i} \to h_ U \]

becomes surjective after sheafification.

Proof. By Lemma 7.11.2 above we have to show that $\coprod \nolimits _{i \in I} h_{U_ i}^\# \to h_ U^\# $ is an epimorphism. Let $\mathcal{F}$ be a sheaf of sets. A morphism $h_ U^\# \to \mathcal{F}$ corresponds to a section $s \in \mathcal{F}(U)$. Hence the injectivity of $\mathop{\mathrm{Mor}}\nolimits (h_ U^\# , \mathcal{F}) \to \prod _ i \mathop{\mathrm{Mor}}\nolimits (h_{U_ i}^\# , \mathcal{F})$ follows directly from the sheaf property of $\mathcal{F}$. $\square$

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