Lemma 7.22.4. Let $\mathcal{C}$ and $\mathcal{D}$ be sites and let $v : \mathcal{D} \to \mathcal{C}$ be a continuous functor. Assume $v$ has a left adjoint $u : \mathcal{C} \to \mathcal{D}$. Then
In particular, $v$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.
Proof. Let $U$ be an object of $\mathcal{C}$ and let $\{ V_ j \to u(U)\} _{j \in J}$ be a covering in $\mathcal{D}$. Then $\{ v(V_ j) \to v(u(U))\} _{i \in I}$ is a covering in $\mathcal{C}$. Via the adjunction map $U \to v(u(U))$ we can base change this to a covering $\{ W_ j \to U\} _{j \in J}$ with $W_ j = v(V_ j) \times _{v(u(U))} U$. Denoting $p_ j : W_ j \to v(V_ j)$ the first projection, we obtain maps
where the second arrow is the adjunction map. This determines a morphism $\{ u(W_ j) \to u(U)\} \to \{ V_ j \to u(U)\} $ of families of maps with fixed target, showing that $u$ is indeed cocontinuous. The other statements are immediate from Lemmas 7.22.1 and 7.22.2. $\square$
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