Kerodon

Example 4.8.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then the functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is full (in the sense of Definition 4.8.5.1) if and only if the functor $F$ is full (in the usual category-theoretic sense). Similarly, $\operatorname{N}_{\bullet }(F)$ is faithful (in the sense of Definition 4.8.5.2) if and only if $F$ is faithful. Consequently, we can view Definitions 4.8.5.1 and Definition 4.8.5.2 as generalizations of their classical counterparts.