Theorem 9.5.2.1 (Adjoint Functor Theorem). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then:
- $(1)$
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it preserves small colimits.
- $(2)$
A functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if it is accessible and preserves small limits.