Variant 11.5.0.107. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor. The following conditions are equivalent:
The functor $U$ is an opfibration in groupoids (Definition 4.2.2.1).
The functor $U$ is a cocartesian fibration and every morphism of $\operatorname{\mathcal{E}}$ is $U$-cocartesian.
The functor $U$ is a cocartesian 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.