Lemma 7.32.10. Let $\mathcal{C}$ be a site. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. The functor $p_* : \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorphisms.

**Proof.**
Because $p_*$ is a right adjoint it commutes with arbitrary limits and it is left exact. The fact that $p^{-1}p_*E \to E$ is a canonically split surjection implies that $p_*$ is faithful, reflects injections, reflects surjections, and reflects isomorphisms. By Lemma 7.32.7 we may assume that $p$ comes from a point $u : \mathcal{C} \to \textit{Sets}$ of the underlying site $\mathcal{C}$. In this case the sheaf $p_*E$ is given by

see Equation (7.32.3.1) and Definition 7.32.6. It follows immediately from this formula that $p_*$ transforms surjections into surjections and coequalizers into coequalizers. $\square$

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