The Stacks project

Lemma 7.14.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume all the categories $(\mathcal{I}_ V^ u)^{opp}$ of Section 7.5 are filtered. Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_ s$ is exact.

Proof. Since $u_ s$ is the left adjoint of $u^ s$ we see that $u_ s$ is right exact, see Categories, Lemma 4.24.6. Hence it suffices to show that $u_ s$ is left exact. In other words we have to show that $u_ s$ commutes with finite limits. Because the categories $\mathcal{I}_ Y^{opp}$ are filtered we see that $u_ p$ commutes with finite limits, see Categories, Lemma 4.19.2 (this also uses the description of limits in $\textit{PSh}$, see Section 7.4). And since sheafification commutes with finite limits as well (Lemma 7.10.14) we conclude because $u_ s = \# \circ u_ p$. $\square$

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