Proposition 7.14.7. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume furthermore the following:

1. the category $\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\mathcal{D}$ , and

2. the category $\mathcal{C}$ has fibre products and $u$ commutes with them.

Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_ s$ is exact.

Proof. This follows from Lemmas 7.5.2 and 7.14.6. $\square$

Comment #993 by on

Suggested slogan: Continuous functors preserving final objects and fibre products induce morphisms of sites.

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