Kerodon

Remark 9.5.3.4. The notation of Construction 9.5.3.1 and Variant 9.5.3.2 is motivated by the adjoint functor theorem (Theorem 9.5.2.1): if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are presentable $\infty $-categories, morphisms from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ are functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which are left adjoints, and morphisms from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ are functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which are right adjoints.