Remarks 8.3.2. Two remarks on Definition 8.3.1 are in order. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\{ f_ i : U_ i \to U\} _{i \in I}$, and $(X_ i, \varphi _{ij})$ be as in Definition 8.3.1.

1. There is a diagonal morphism $\Delta : U_ i \to U_ i \times _ U U_ i$. We can pull back $\varphi _{ii}$ via this morphism to get an automorphism $\Delta ^\ast \varphi _{ii} \in \text{Aut}_{U_ i}(X_ i)$. On pulling back the cocycle condition for the triple $(i, i, i)$ by $\Delta _{123} : U_ i \to U_ i \times _ U U_ i \times _ U U_ i$ we deduce that $\Delta ^\ast \varphi _{ii} \circ \Delta ^\ast \varphi _{ii} = \Delta ^\ast \varphi _{ii}$; thus $\Delta ^\ast \varphi _{ii} = \text{id}_{X_ i}$.

2. There is a morphism $\Delta _{13}: U_ i \times _ U U_ j \to U_ i \times _ U U_ j \times _ U U_ i$ and we can pull back the cocycle condition for the triple $(i, j, i)$ to get the identity $(\sigma ^\ast \varphi _{ji}) \circ \varphi _{ij} = \text{id}_{\text{pr}_0^\ast X_ i}$, where $\sigma : U_ i \times _ U U_ j \to U_ j \times _ U U_ i$ is the switching morphism.

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