Kerodon

Remark 1.5.2.7. In the situation of Proposition 1.5.2.6, an arbitrary morphism of simplicial sets $\sigma : K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ 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 1.3.7.5). 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 $F$ on a path $p$ depends only the endpoints of $p$.