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Example Let $G$ be the directed graph depicted in the diagram

\[ \xymatrix { \bullet \ar [r] & \bullet \ar [r] & \bullet . } \]

Then the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ can be identified with the inclusion of simplicial sets $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. In this case, Theorem reduces to the statement that the map

\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Lambda ^{2}_{1}, \operatorname{\mathcal{C}}) \]

is a trivial Kan fibration, which is equivalent to the assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category by virtue of Theorem