The Stacks project

Proof. Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and let $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the full subcategory consisting of those $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism. By Lemma 7.43.5 the functor $i_*$ is isomorphic to the inclusion functor $\iota : E \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Let $j_{\mathcal{F}} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the localization functor (Lemma 7.30.1). Note that $E$ can also be described as the collection of sheaves $\mathcal{G}$ such that $j_\mathcal {F}^{-1}\mathcal{G} = *$.

Let $a, b : \mathcal{G}_1 \to \mathcal{G}_2$ be two morphism of $E$. To prove $\iota$ commutes with coequalizers it suffices to show that the coequalizer of $a$, $b$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ lies in $E$. This is clear because the coequalizer of two morphisms $* \to *$ is $*$ and because $j_\mathcal {F}^{-1}$ is exact. Similarly for pushouts.

Thus $i_*$ satisfies properties (5), (6), and (7) of Lemma 7.41.1 and hence the morphism $i$ satisfies all properties mentioned in that lemma, in particular the ones mentioned in this lemma. $\square$