The Stacks project

Lemma 8.6.8. Let $\mathcal{C}$ be a site. Let $\mathcal{S}$ be a stack in groupoids over $\mathcal{C}$ and let $\mathcal{S}_ i$, $i = 1, 2$ be stacks in setoids over $\mathcal{C}$. Let $f_ i : \mathcal{S}_ i \to \mathcal{S}$ be $1$-morphisms. Then the $2$-fibre product

\[ \mathcal{S}_1 \times _{f_1, \mathcal{S}, f_2} \mathcal{S}_2 \]

is a stack in setoids over $\mathcal{C}$.

Proof. This is a special case of Lemma 8.6.7 as $f_2$ is faithful. $\square$

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