The Stacks project

Remark 4.35.8. Let $\mathcal{C}$ be a category. Let $f : \mathcal{X} \to \mathcal{S}$ and $g : \mathcal{Y} \to \mathcal{S}$ be $1$-morphisms of categories fibred in groupoids over $\mathcal{C}$. Let $p : \mathcal{S} \to \mathcal{C}$ be the given functor. We claim the $2$-fibre product of Lemma 4.35.7 is canonically equivalent (as a category) to the one in Example 4.31.3. Objects of the former are quadruples $(U, x, y, \alpha )$ where $p(\alpha ) = \text{id}_ U$ (see Lemma 4.32.3) and objects of the latter are triples $(x, y, \alpha )$ (see Example 4.31.3). The equivalence between the two categories is given by the rules $(U, x, y, \alpha ) \mapsto (x, y, \alpha )$ and $(x, y, \alpha ) \mapsto (p(f(x)), x, y', \alpha ')$ where $\alpha ' = g(\gamma )^{-1} \circ \alpha $ and $\gamma : y' \to y$ is a lift of the arrow $p(\alpha ) : p(f(x)) \to p(g(y))$. Details omitted.