Lemma 7.10.3. The constructions above define a presheaf $\mathcal{F}^+$ together with a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$.

Proof. All we have to do is to show that given morphisms $W \to V \to U$ the composition $\mathcal{F}^+(U) \to \mathcal{F}^+(V) \to \mathcal{F}^+(W)$ equals the map $\mathcal{F}^+(U) \to \mathcal{F}^+(W)$. This can be shown directly by verifying that, given a covering $\{ U_ i \to U\}$ and $s = (s_ i) \in H^0(\{ U_ i \to U\} , \mathcal{F})$, we have canonically $W \times _ U U_ i \cong W \times _ V (V \times _ U U_ i)$, and $s_ i|_{W \times _ U U_ i}$ corresponds to $(s_ i|_{V \times _ U U_ i})|_{W \times _ V (V \times _ U U_ i)}$ via this isomorphism. $\square$

Comment #8568 by Alejandro González Nevado on

SS: The colimit, when the coverings vary over the opposite of the category of all coverings of a fixed object on a site, of the zeroth Čech cohomology of a fixed presheaf of sets with respect to these coverings of the fixed object on the site defines a presheaf (the zeroth Čech cohomology of the fixed presheaf of sets over the fixed object on the site) and a canonical map of presheaves from the original sheaf to the one produced here via the colimit.

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