Example 4.8.0.4. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ be a functor of ordinary categories and let $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}_0)$ denote the induced of $\infty $-categories. Then:
The functor $F$ is $0$-full if and only if $F_0$ is essentially surjective.
The functor $F$ is $1$-full if and only if $F_0$ is full.
The functor $F$ is $2$-full if and only if $F_0$ is faithful.
For $m \geq 3$, the functor $F$ is automatically $m$-full (see Exercise 1.3.1.5).