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

Suppose all finite limits exist in $\mathcal{A}$. We say $F$ is

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

*right exact*if it commutes with all finite colimits.We say $F$ is

*exact*if it is both left and right exact.

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