The Stacks project

Lemma 4.35.12. Let $\mathcal{C}$ be a category. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. If $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids and $p$ factors through $p' : \mathcal{S} \to \mathcal{C}/U$ then $p' : \mathcal{S} \to \mathcal{C}/U$ is fibred in groupoids.

Proof. We have already seen in Lemma 4.33.11 that $p'$ is a fibred category. Hence it suffices to prove the fibre categories are groupoids, see Lemma 4.35.2. For $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have

\[ \mathcal{S}_ V = \coprod \nolimits _{f : V \to U} \mathcal{S}_{(f : V \to U)} \]

where the left hand side is the fibre category of $p$ and the right hand side is the disjoint union of the fibre categories of $p'$. Hence the result. $\square$


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