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Remark In the situation of Proposition, an arbitrary map of simplicial sets $\sigma : K_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with a functor $F: \operatorname{Path}[G] \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{Path}[G]$ denotes the path category of the graph $G$ (Proposition The commutativity of the diagram $\sigma $ is equivalent to the requirement that $F$ factors through the quotient functor $\operatorname{Path}[G] \twoheadrightarrow \operatorname{Vert}(G)$: that is, the value of the functor $F$ on a path depends only the endpoints of that path.