The Stacks project

Lemma 7.38.6. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $\{ (q_ i, v_ i)\} _{i\in I}$ be a conservative family of points of $\mathcal{D}$. If each functor $u_ i = v_ i \circ u$ defines a point of $\mathcal{C}$, then $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.

Proof. Denote $p_ i$ the stalk functor ( on $\textit{PSh}(\mathcal{C})$ corresponding to the functor $u_ i$. We have

\[ (f^{-1}\mathcal{F})_{q_ i} = (u_ s\mathcal{F})_{q_ i} = (u_ p\mathcal{F})_{q_ i} = \mathcal{F}_{p_ i} \]

The first equality since $f^{-1} = u_ s$, the second equality by Lemma 7.32.5, and the third by Lemma 7.34.1. Hence if $p_ i$ is a point, then pulling back by $f$ and then taking stalks at $q_ i$ is an exact functor. Since the family of points $\{ q_ i\} $ is conservative, this implies that $f^{-1}$ is an exact functor and we see that $f$ is a morphism of sites by Definition 7.14.1. $\square$

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