The Stacks project

Lemma 8.6.7. Let $\mathcal{C}$ be a site. Let $\mathcal{S}, \mathcal{T}$ be stacks in groupoids over $\mathcal{C}$ and let $\mathcal{R}$ be a stack in setoids over $\mathcal{C}$. Let $f : \mathcal{T} \to \mathcal{S}$ and $g : \mathcal{R} \to \mathcal{S}$ be $1$-morphisms. If $f$ is faithful, then the $2$-fibre product

\[ \mathcal{T} \times _{f, \mathcal{S}, g} \mathcal{R} \]

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

Proof. Immediate from the explicit description of the $2$-fibre product in Categories, Lemma 4.32.3. $\square$

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