The Stacks project

Lemma 7.39.4. Let $\mathcal{C}$ be a site. Let $I$ be a set and for $i \in I$ let $U_ i$ be an object of $\mathcal{C}$ such that

  1. $\coprod h_{U_ i}$ surjects onto the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

  2. $\mathcal{C}/U_ i$ satisfies the hypotheses of Proposition 7.39.3.

Then $\mathcal{C}$ has enough points.

Proof. By assumption (2) and the proposition $\mathcal{C}/U_ i$ has enough points. The points of $\mathcal{C}/U_ i$ give points of $\mathcal{C}$ via the procedure of Lemma 7.34.2. Thus it suffices to show: if $\phi : \mathcal{F} \to \mathcal{G}$ is a map of sheaves on $\mathcal{C}$ such that $\phi |_{\mathcal{C}/U_ i}$ is an isomorphism for all $i$, then $\phi $ is an isomorphism. By assumption (1) for every object $W$ of $\mathcal{C}$ there is a covering $\{ W_ j \to W\} _{j \in J}$ such that for $j \in J$ there is an $i \in I$ and a morphism $f_ j : W_ j \to U_ i$. Then the maps $\mathcal{F}(W_ j) \to \mathcal{G}(W_ j)$ are bijective and similarly for $\mathcal{F}(W_ j \times _ W W_{j'}) \to \mathcal{G}(W_ j \times _ W W_{j'})$. The sheaf condition tells us that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective as desired. $\square$

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