Definition 1.2.5.1. Let $\operatorname{\mathcal{C}}$ be a category. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ if, for every category $\operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]
is bijective (note that the map on the left is always bijective, by virtue of Proposition 1.2.2.1).