Kerodon

Remark 9.2.5.20. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $(\kappa ,\lambda )$-cocomplete, and let $C$ be a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. If $F$ admits a $(\kappa ,\lambda )$-finitary right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then $F(C)$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{D}}$. This follows from the observation that $F(C)$ corepresents the composite functor $\operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\xrightarrow {\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet )} \operatorname{\mathcal{S}}_{< \mu }$, where $\mu $ is any regular cardinal for which $\operatorname{\mathcal{C}}$ is locally $\mu $-small. See Remark 6.2.6.3.