Theorem 1.5.7.1. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the map of simplicial sets $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ induces a trivial Kan fibration of simplicial sets $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 1.5.7.1. Let $G$ be a graph and let $\operatorname{\mathcal{C}}$ be an $\infty $-category; we wish to show that the canonical map
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}}) \]
is a trivial Kan fibration. This follows from Proposition 1.5.7.6, since the inclusion $G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne (Proposition 1.5.7.3). $\square$