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 (7.32.1.1) 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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