Lemma 7.42.5. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Then $u^ s = u^ p : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ commutes with pushouts and coequalizers (and more generally finite connected colimits).

Proof. Let $\mathcal{I}$ be a finite connected index category. Let $\mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $i \mapsto \mathcal{G}_ i$ by a diagram. We know that the colimit of this diagram is the sheafification of the colimit in the category of presheaves, see Lemma 7.10.13. Denote $\mathop{\mathrm{colim}}\nolimits ^{Psh}$ the colimit in the category of presheaves. Since $\mathcal{I}$ is finite and connected we see that $\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i$ is a presheaf satisfying the assumptions of Lemma 7.42.4 (because a finite connected colimit of singleton sets is a singleton). Hence that lemma gives

\begin{align*} u^ s(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i) & = u^ s((\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i)^\# ) \\ & = (u^ p(\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i))^\# \\ & = (\mathop{\mathrm{colim}}\nolimits ^{PSh}_ i u^ p(\mathcal{G}_ i))^\# \\ & = \mathop{\mathrm{colim}}\nolimits _ i u^ s(\mathcal{G}_ i) \end{align*}

as desired. $\square$

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