Theorem 59.13.2. Let \mathcal{C} be a site and \mathcal{F} a presheaf on \mathcal{C}.
The rule
U \mapsto \mathcal{F}^+(U) := \mathop{\mathrm{colim}}\nolimits _{\mathcal{U} \text{ covering of }U} \check H^0(\mathcal{U}, \mathcal{F})is a presheaf. And the colimit is a directed one.
There is a canonical map of presheaves \mathcal{F} \to \mathcal{F}^+.
If \mathcal{F} is a separated presheaf then \mathcal{F}^+ is a sheaf and the map in (2) is injective.
\mathcal{F}^+ is a separated presheaf.
\mathcal{F}^\# = (\mathcal{F}^+)^+ is a sheaf, and the canonical map induces a functorial isomorphism
\mathop{\mathrm{Hom}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}^\# , \mathcal{G})for any \mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C}).
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