Corollary 2.4.4.4. The homotopy coherent nerve functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint
\[ \operatorname{Path}[ - ]_{\bullet }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat_{\Delta }}, \]
which associates to each simplicial set $S_{\bullet }$ the path category $\operatorname{Path}[ S ]_{\bullet }$ of Notation 2.4.4.2.