Remark 8.6.8.17. The formation-of-adjoints functor $\Psi : \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$ induces a functor of homotopy categories $\mathrm{h} \mathit{\Psi }: \mathrm{h} \mathit{ \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}_{\operatorname{RAd}}} \subset \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}$, which is a homotopy transport representation for the cartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}^{0} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$ (Corollary 8.6.8.10). By virtue of Proposition 6.2.3.6, this functor can be described more concretely as follows:
The functor $\mathrm{h} \mathit{\Psi }$ carries every small $\infty $-category $\operatorname{\mathcal{C}}$ (viewed as an object of $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$) to itself (viewed as an object of $\operatorname{\mathcal{QC}}_{\operatorname{RAd}}$).
If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor between small $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then the functor $\mathrm{h} \mathit{\Psi }$ carries the isomorphism class $[F]$ (viewed as a morphism in the homotopy cateogry $\mathrm{h} \mathit{\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}}$) to the isomorphism class $[G]$ (viewed as a morphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{\mathcal{QC}}_{\operatorname{RAd}} }$.