Definition 1.3.6.1. Let $\operatorname{\mathcal{C}}$ be a category. We will say that a map of simplicial sets $u: S \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}$ as the homotopy category of $S$ if, for every category $\operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]
is bijective (note that the map on the left is always bijective, by virtue of Proposition 1.3.3.1).