Remark 9.5.3.10. The equivalence $\Psi : (\operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ of Corollary 9.5.3.9 can be described more informally as follows:
It carries each presentable $\infty $-category $\operatorname{\mathcal{C}}$ (regarded as an object of $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) to itself (regarded as an object of $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$).
It carries each cocontinuous functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (regarded as a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) to its right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (regarded as a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$).
More precisely, this description characterizes the equivalence of homotopy categories induced by the functor $\Psi $ (see Remark 8.6.8.20).