The Stacks project

Definition 4.23.1. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor.

1. Suppose all finite limits exist in $\mathcal{A}$. We say $F$ is left exact if it commutes with all finite limits.

2. Suppose all finite colimits exist in $\mathcal{A}$. We say $F$ is right exact if it commutes with all finite colimits.

3. We say $F$ is exact if it is both left and right exact.