Example 7.1.4.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
The functor $U$ is fully faithful.
Every object of $\operatorname{\mathcal{C}}$ is $U$-initial.
Every object of $\operatorname{\mathcal{C}}$ is $U$-final.