The Stacks project

Lemma 4.35.7. Let $\mathcal{C}$ be a category. The $2$-category of categories fibred in groupoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Lemma 4.32.3.

Proof. By Lemma 4.33.10 the fibre product as described in Lemma 4.32.3 is a fibred category. Hence it suffices to prove that the fibre categories are groupoids, see Lemma 4.35.2. By Lemma 4.32.5 it is enough to show that the $2$-fibre product of groupoids is a groupoid, which is clear (from the construction in Lemma 4.31.4 for example). $\square$

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