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 $\mathcal{F}$.

Proof. The first assertion is immediate from the definition of a sheaf, assumptions (1) and (2), and the definition of $u^ pE$. The second is a restatement of the adjointness of $u^ p$ and the stalk functor (Lemma 7.32.4) restricted to sheaves. The third assertion follows as, for any set $E$, we have

$\text{Map}(\mathcal{F}_ p, E) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^ pE) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}^\# , u^ sE) = \text{Map}(\mathcal{F}^\# _ p, E)$

by the adjointness property of sheafification. $\square$

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