The Stacks project

Lemma 7.22.2. Notation and assumptions as in Lemma 7.22.1. If in addition $v$ is continuous then $v$ defines a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ whose associated morphism of topoi is equal to $g$.

Proof. We will use the results of Lemma 7.22.1 without further mention. To prove that $v$ defines a morphism of sites $f$ as in the statement of the lemma, we have to show that $v_ s$ is an exact functor (see Definition 7.14.1). Since $v_ s\mathcal{G} = (v_ p\mathcal{G})^\# = g^{-1}\mathcal{G}$ this follows from the fact that $g$ is a morphism of topoi. Then we see that $f^{-1} = v_ s = g^{-1}$ and we find that $f = g$ as morphisms of topoi. $\square$

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