Example 4.6.2.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is fully faithful if and only if the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is fully faithful (in the sense of Definition 4.6.2.1). Consequently, we can regard Definition 4.6.2.1 as a generalization of the classical notion of fully faithful functor.
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