The Stacks project

Lemma 7.21.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that

  1. $u$ is cocontinuous,

  2. $u$ is continuous, and

  3. fibre products and equalizers exist in $\mathcal{C}$ and $u$ commutes with them.

In this case the functor $g_!$ above commutes with fibre products and equalizers (and more generally with finite connected limits).

Proof. Assume (a), (b), and (c). We have $g_! = (u_ p\ )^\# $. Recall (Lemma 7.10.1) that limits of sheaves are equal to the corresponding limits as presheaves. And sheafification commutes with finite limits (Lemma 7.10.14). Thus it suffices to show that $u_ p$ commutes with fibre products and equalizers. To do this it suffices that colimits over the categories $(\mathcal{I}_ V^ u)^{opp}$ of Section 7.5 commute with fibre products and equalizers. This follows from Lemma 7.5.1 and Categories, Lemma 4.19.9. $\square$

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