Kerodon

Corollary 9.2.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite limits. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left exact (in the sense of Definition 9.2.5.1) if and only if the functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is left exact (in the sense of Definition 9.2.5.5).