Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 1.4.2.7. In the situation of Proposition 1.4.2.6, 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 1.2.6.5). 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 the functor $F$ on a path depends only the endpoints of that path.