Remark 5.5.6.15. The statement of Proposition 5.5.6.14 can be made more precise: Theorem 5.4.9.2 supplies an explicit $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched isomorphism from the identity functor $\operatorname{id}_{ \mathrm{h} \mathit{ \operatorname{QCat}} }$ to the composition
\[ \mathrm{h} \mathit{\operatorname{QCat}} \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \xrightarrow { \operatorname{hTr}_{ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}} } \mathrm{h} \mathit{\operatorname{QCat}}, \]
which carries each small $\infty $-category $\operatorname{\mathcal{C}}$ to the equivalence
\[ \theta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \{ \operatorname{\mathcal{C}}\} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} = \operatorname{hTr}_{ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}(\operatorname{\mathcal{C}}) \]
described in Remark 5.5.6.13.