The Stacks project

Lemma 7.22.2. In the situation of Lemma 7.22.1. We have $g_* = v^ s = v^ p$ and $g^{-1} = v_ s = (v_ p\ )^\# $. If $v$ is continuous then $v$ defines a morphism of sites $f$ from $\mathcal{C}$ to $\mathcal{D}$ whose associated morphism of topoi is equal to the morphism $g$ associated to the cocontinuous functor $u$. In other words, a continuous functor which has a cocontinuous left adjoint defines a morphism of sites.

Proof. Clear from the discussion above the lemma and Definitions 7.14.1 and Lemma 7.15.2. $\square$

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