$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 9.5.2.5. Let $\kappa < \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $(\kappa ,\lambda )$-compactly generated, and suppose that the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it satisfies the following pair of conditions:
- $(1)$
The functor $F$ is $\lambda $-cocontinuous.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}_{< \kappa }$ and every object $D \in \operatorname{\mathcal{D}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D )$ is essentially $\lambda $-small.
Proof.
Fix an uncountable regular cardinal $\mu $ such that $\operatorname{\mathcal{D}}$ is locally $\mu $-small, so that every object $D \in \operatorname{\mathcal{D}}$ determines a representable functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$. By virtue of Corollary 6.2.6.2, the functor $F$ admits a right adjoint if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the composition $\operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow {F^{\operatorname{op}}} \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { h_{D} } \operatorname{\mathcal{S}}_{< \mu }$ is representable by an object of $\operatorname{\mathcal{C}}$. The equivalence of this condition with $(1)$ and $(2)$ now follows from Proposition 9.5.1.11 (together with Proposition 7.4.1.22).
$\square$