Kerodon

Variant 9.2.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. We say that $F$ is right exact if, for every right fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is filtered, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also filtered. Equivalently, $F$ is right exact if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is left exact.