Example 4.8.7.2. For small values of $n$, we can make Definition 4.8.7.1 more concrete:
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $1$-connective if and only if it is full and essentially surjective.
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $0$-connective if and only if it is essentially surjective.
For $n < 0$, every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $n$-connective.