Kerodon

Variant 9.4.2.2. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be a $\kappa $-compactly generated $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-compact if it is $\kappa $-finitary (Definition 9.2.2.3) and carries $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ to $\kappa $-compact objects of $\operatorname{\mathcal{D}}$. We let $\operatorname{Fun}^{\kappa -\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $\kappa $-compact functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.