Kerodon

Remark 9.4.2.13 (Retracts). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be $(\kappa ,\lambda )$-compact functor. If $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a retract of $F$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), then $G$ is also $(\kappa ,\lambda )$-compact (see Remark 9.2.5.18). In particular, if $F$ and $G$ are isomorphic, then $F$ is $(\kappa ,\lambda )$-compact if and only if $G$ is $(\kappa ,\lambda )$-compact.