The Stacks project

Lemma 7.32.5. Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that for every covering $\{ U_ i \to U\} $ of $\mathcal{C}$

1. the map $\coprod u(U_ i) \to u(U)$ is surjective, and

2. the maps $u(U_ i \times _ U U_ j) \to u(U_ i) \times _{u(U)} u(U_ j)$ are surjective.

Then we have

1. the presheaf $u^ pE$ is a sheaf for all sets $E$, denote it $u^ sE$,

2. the stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$ and the functor $u^ s: \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ are adjoint, and

3. we have $\mathcal{F}_ p = \mathcal{F}^\# _ p$ for every presheaf of sets $\mathcal{F}$.