Construction 4.6.8.23. Let $K$ be a simplicial set, let $\operatorname{\mathcal{E}}[K]$ be the simplicial category of Notation 4.6.8.1, and let
\[ \theta : K = \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}(x,y)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}[K])}( x, y) \]
be the comparison map of Construction 4.6.8.3. We let $\rho _{K}: \Phi (K) \rightarrow K$ denote the image of $\theta $ under the bijection of Corollary 4.6.8.19.