Exercise 5.0.0.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Show that the following conditions are equivalent:
The functor $U$ is a fibration in groupoids (Definition 4.2.2.1).
The functor $U$ is a cartesian fibration and every morphism of $\operatorname{\mathcal{E}}$ is $U$-cartesian.
The functor $U$ is a cartesian fibration and, for every object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a groupoid.
For a more general statement, see Proposition 5.1.4.15.