Kerodon

Construction 8.6.8.16. Let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the $\infty $-category introduced in Definition 5.5.6.10 (whose objects are given by pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a small $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object). Let $U: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ be the forgetful functor, given on objects by the construction $(\operatorname{\mathcal{C}}, C) \mapsto \operatorname{\mathcal{C}}$. We let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}^{0} \subseteq \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the inverse image of the subcategory $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}$. It follows from Remark 8.6.8.15 that $U$ restricts to a forgetful functor $U^{0}: \operatorname{\mathcal{QC}}_{\operatorname{Obj}}^{0} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$ which is both a cartesian fibration and a cocartesian fibration. Applying Proposition 8.6.8.3 (and Remark 8.6.8.15), we deduce that $U^{0}$ admits a contravariant transport representation $\Psi : \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$, which is uniquely determined up to isomorphism. We will refer to $\Psi $ as the formation-of-adjoints functor.