Remark 4.8.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
The functor $F$ is faithful if and only if it is $n$-full for each $n \geq 2$ (see Example 3.5.9.3).
The functor $F$ is fully faithful if and only if it is $n$-full for each $n \geq 1$ (see Remark 4.8.5.3).
The functor $F$ is an equivalence of $\infty $-categories if and only if it is $n$-full for each $n \geq 0$ (see Theorem 4.6.2.21).