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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