Remark 4.5.1.7. The right adjoint $\mathrm{h} \mathit{\operatorname{QCat}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ of Corollary 4.5.1.6 can be described more explicitly as follows:
To each $\infty $-category $\operatorname{\mathcal{C}}$, it associates the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ of Construction 4.4.3.1.
To each morphism $[F]: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ (given by the isomorphism class of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$), it associates the homotopy class $[F^{\simeq }]$ of the underlying map of cores $F^{\simeq } = F|_{ \operatorname{\mathcal{C}}^{\simeq } }$ (note that the homotopy class of $F^{\simeq }$ depends only on the isomorphism class of $F$, by virtue of Remark 4.4.4.5).