Example 9.4.2.17. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated and $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-cocomplete. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then:
The functor $F$ is automatically $(\kappa ,\lambda )$-finitary (Corollary 7.1.4.28).
If $G$ is $(\kappa ,\lambda )$-finitary, then $F$ is $(\kappa ,\lambda )$-compact (Remark 9.2.5.20).
For a partial converse, see Proposition 