Lemma 7.38.5. Let $\mathcal{C}$ be a site. Let $\{ U_ i\} _{i \in I}$ be a family of objects of $\mathcal{C}$. Assume

1. $\coprod h_{U_ i}^\# \to *$ is a surjective map of sheaves, and

2. each localization $\mathcal{C}/U_ i$ has enough points.

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

Proof. For each $i \in I$ let $\{ p_ j\} _{j \in J_ i}$ be a conservative family of points of $\mathcal{C}/U_ i$. For $j \in J_ i$ denote $q_ j : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ the composition of $p_ j$ with the localization morphism $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then $q_ j$ is a point, see Lemma 7.34.3. We claim that the family of points $\{ q_ j\} _{j \in \coprod J_ i}$ is conservative. Namely, let $\mathcal{F} \to \mathcal{G}$ be a map of sheaves on $\mathcal{C}$ such that $\mathcal{F}_{q_ j} \to \mathcal{G}_{q_ j}$ is an isomorphism for all $j \in \coprod J_ i$. Let $W$ be an object of $\mathcal{C}$. By assumption (1) there exists a covering $\{ W_ a \to W\}$ and morphisms $W_ a \to U_{i(a)}$. Since $(\mathcal{F}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{F}_{q_ j}$ and $(\mathcal{G}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{G}_{q_ j}$ by Lemma 7.34.3 we see that $\mathcal{F}|_{U_{i(a)}} \to \mathcal{G}|_{U_{i(a)}}$ is an isomorphism since the family of points $\{ p_ j\} _{j \in J_{i(a)}}$ is conservative. Hence $\mathcal{F}(W_ a) \to \mathcal{G}(W_ a)$ is bijective for each $a$. Similarly $\mathcal{F}(W_ a \times _ W W_ b) \to \mathcal{G}(W_ a \times _ W W_ b)$ is bijective for each $a, b$. By the sheaf condition this shows that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective, i.e., $\mathcal{F} \to \mathcal{G}$ is an isomorphism. $\square$

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