Kerodon

Remark 9.2.2.24 (Retracts). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits, and let $C \in \operatorname{\mathcal{C}}$ be an object. If $C$ is $(\kappa ,\lambda )$-compact, then any retract of $C$ is also $(\kappa ,\lambda )$-compact. This follows from Corollary 8.5.1.14. If $\kappa $ is uncountable, then it can also be deduced from Proposition 9.2.2.21, since any retract of $C$ can be realized as the colimit of a diagram $C \xrightarrow {e} C \xrightarrow {e} C \xrightarrow {e} C \rightarrow \cdots $; see Proposition 8.5.4.17.