The Stacks project

Lemma 7.19.4. A continuous functor of sites which has a continuous left adjoint defines a morphism of sites.

Proof. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $w : \mathcal{D} \to \mathcal{C}$ be a continuous left adjoint. Then $u_ p = w^ p$ by Lemma 7.19.3. Hence $u_ s = w^ s$ has a left adjoint, namely $w_ s$ (Lemma 7.13.3). Thus $u_ s$ has both a right and a left adjoint, whence is exact (Categories, Lemma 4.24.6). $\square$

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