The Stacks project

Lemma 8.6.3. Let $\mathcal{C}$ be a site. Let $\mathcal{S}$ be a category fibred in setoids over $\mathcal{C}$. Then $\mathcal{S}$ is a stack in setoids if and only if the unique equivalent category $\mathcal{S}'$ fibred in sets (see Categories, Lemma 4.39.5) is a stack in sets. In other words, if and only if the presheaf

\[ U \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)/\! \! \cong \]

is a sheaf.

Proof. Omitted. $\square$

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