Kerodon

Remark 9.2.2.13 (Retracts). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete, and let $\operatorname{\mathcal{D}}$ be another $\infty $-category. Suppose we are given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $F$ is a retract of $G$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). If $G$ is $(\kappa ,\lambda )$-finitary, then $F$ is also $(\kappa ,\lambda )$-finitary (see Corollary 8.5.1.12). In particular, the condition that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-finitary depends only on the isomorphism class of $F$.