Kerodon

Definition 9.2.5.10. Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is $\kappa $-left exact if, for every left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is $\kappa $-cofiltered, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also $\kappa $-cofiltered (see Definition 9.1.1.4). We say that $F$ is $\kappa $-right exact if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $\kappa $-left exact. We let $\operatorname{Fun}^{\kappa -\operatorname{lex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are $\kappa $-left exact, and $\operatorname{Fun}^{\kappa -\operatorname{rex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ the full subcategory spanned by those functors which are $\kappa $-right exact.