The Stacks project

Lemma 7.15.2. Given a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ corresponding to the functor $u : \mathcal{C} \to \mathcal{D}$ the pair of functors $(f^{-1} = u_ s, f_* = u^ s)$ is a morphism of topoi.

Proof. This is obvious from Definition 7.14.1. $\square$

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