Proposition 8.6.8.19. The formation-of-adjoints functor $\Psi : \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$ is an equivalence of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We wish to show that composition with $\Psi $ induces a bijection
\[ \{ \textnormal{Diagrams $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$} \} / \textnormal{Isomorphism} \rightarrow \{ \textnormal{Diagrams $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$} \} / \textnormal{Isomorphism}. \]
Using Theorem 5.6.0.2 and Remark 8.6.8.18, we can identify the left hand side with the set of equivalence classes of (essentially small) cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which are also cartesian fibrations. Under this identification, the map is given by the formation of contravariant transport representations, and is therefore bijective by virtue of Proposition 8.6.8.3 (and Remark 8.6.8.17). $\square$