Remark 1.5.7.4. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an ordinary category. Combining Proposition 1.5.7.3 with Variant 1.5.6.8, we deduce that the canonical map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \]
is bijective. Combining this observation with Proposition 1.3.3.1, we obtain a bijection
\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]
Allowing $\operatorname{\mathcal{C}}$ to vary, we recover the assertion that $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G] )$ exhibits $\operatorname{Path}[G]$ as the homotopy category of $G_{\bullet }$ (Proposition 1.3.7.5).