Notation 8.6.8.14. Let $\operatorname{\mathcal{QC}}$ be the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1)). We define (non-full) subcategories $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}\supset \operatorname{\mathcal{QC}}_{\operatorname{RAd}}$ as follows:
Every object of $\operatorname{\mathcal{QC}}$ belongs to the subcategories $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$ and $\operatorname{\mathcal{QC}}_{\operatorname{RAd}}$.
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between (small) $\infty $-categories, which we regard as a morphism in $\operatorname{\mathcal{QC}}$. Then $F$ belongs to $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$ if and only if $F$ is a left adjoint: that is, there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is right adjoint to $F$. Similarly, $F$ belongs to $\operatorname{\mathcal{QC}}_{\operatorname{RAd}}$ if and only if $F$ is a right adjoint.