Corollary 11.10.2.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. The following conditions are equivalent:
There exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ and an isomorphism of categories $\int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ which carries $U$ to the forgetful functor $\int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.
The functor $U$ is a cartesian fibration.