The Stacks project

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. $\square$