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The Stacks project

Lemma 7.26.5. Let \mathcal{C} be a site. Let \{ U_ i \to U\} _{i \in I} be a covering of \mathcal{C}. The category \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) is equivalent to the category of glueing data via the functor that associates to \mathcal{F} on \mathcal{C}/U the canonical glueing data.

Proof. In Lemma 7.26.1 we saw that the functor is fully faithful, and in Lemma 7.26.4 we proved that it is essentially surjective (by explicitly constructing a quasi-inverse functor). \square


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