Example 8.1.9.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. The following conditions are equivalent:
The morphism $U$ is an isofibration of $\infty $-categories.
Every isomorphism of $\operatorname{\mathcal{C}}$ admits $U$-cocartesian lifts.
Every isomorphism of $\operatorname{\mathcal{C}}$ admits $U$-cartesian lifts.