Example 1.5.1.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. It follows from Proposition 1.3.3.1 that the formation of nerves induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Functors of ordinary categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Functors of $\infty $-categories from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$} \} . } \]
In other words, Definition 1.5.0.1 can be regarded as a generalization of the usual notion of functor to the setting of $\infty $-categories.