Remark 5.5.3.7. The statement of Proposition 5.5.3.6 can be made more precise: Theorem 5.4.9.2 supplies an explicit $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched isomorphism from the identity functor $\operatorname{id}_{\mathrm{h} \mathit{\operatorname{Kan}}}$ to the composition
\[ \mathrm{h} \mathit{\operatorname{Kan}} \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{S}}} \xrightarrow { \operatorname{hTr}_{ \operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}} } \mathrm{h} \mathit{\operatorname{Kan}}, \]
which carries each Kan complex $X$ to the homotopy equivalence $\theta _{X}: X \rightarrow \{ X\} \times _{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\ast } = \operatorname{hTr}_{ \operatorname{\mathcal{S}}_{\ast } / \operatorname{\mathcal{S}}}(X)$ of Remark 5.5.3.5.