The Stacks project

Definition 4.36.2. Let $\mathcal{C}$ be a category. Suppose that $F : \mathcal{C}^{opp} \to \textit{Cat}$ is a functor to the $2$-category of categories. We will write $p_ F : \mathcal{S}_ F \to \mathcal{C}$ for the fibred category constructed in Example 4.36.1. A split fibred category is a fibred category isomorphic (!) over $\mathcal{C}$ to one of these categories $\mathcal{S}_ F$.