Proposition 5.4.9.1. Let $\operatorname{\mathcal{C}}$ be locally Kan simplicial category containing an object $X$, and let $\Phi : \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow {\sim } \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }$ be the isomorphism of Proposition 2.4.6.9. Then the diagram of functors
\[ \xymatrix@R =50pt@C=50pt{ & \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \ar [dl]_{\Phi }^{\sim } \ar [dr]^{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, \bullet )_{\bullet }} & \\ \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} \ar [rr]^-{ \operatorname{hTr}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} / \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }} & & \mathrm{h} \mathit{\operatorname{Kan}} } \]
commutes up to isomorphism.