The Stacks project

Lemma 7.38.4. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. let $\{ (p_ i, u_ i)\} _{i\in I}$ be a family of points of $\mathcal{C}$. For $x \in u_ i(U)$ let $q_{i, x}$ be the point of $\mathcal{C}/U$ constructed in Lemma 7.35.1. If $\{ p_ i\} $ is a conservative family of points, then $\{ q_{i, x}\} _{i \in I, x \in u_ i(U)}$ is a conservative family of points of $\mathcal{C}/U$. In particular, if $\mathcal{C}$ has enough points, then so does every localization $\mathcal{C}/U$.

Proof. We know that $j_{U!}$ induces an equivalence $j_{U!} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $, see Lemma 7.25.4. Moreover, we know that $(j_{U!}\mathcal{G})_{p_ i} = \coprod _ x \mathcal{G}_{q_{i, x}}$, see Lemma 7.35.3. Hence the result follows formally. $\square$

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