Kerodon

Variant 9.4.2.4. 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. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-compact if it is $(\kappa ,\lambda )$-finitary (Definition 9.2.2.6) and carries $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ to $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{D}}$. We let $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $(\kappa ,\lambda )$-compact functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.