# Exact functor

A functor that commutes with finite limits and colimits. More precisely, an additive functor $F : \mathfrak A \rightarrow \mathfrak B$ between Abelian categories $\mathfrak A$ and $\mathfrak B$ is called exact if it maps a short exact sequence

$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$

in $\mathfrak A$ into a short exact sequence

$$0 \rightarrow F ( A) \rightarrow F ( B) \rightarrow F ( C) \rightarrow 0$$

in $\mathfrak B$.

If $\mathfrak A$ and $\mathfrak B$ are non-Abelian categories, then a functor $F : \mathfrak A \rightarrow \mathfrak B$ is sometimes called exact if it maps a commutative diagram

$$A \begin{array}{c} \rightarrow ^ { {\epsilon _ 1} } \\ \mathop \rightarrow \limits _ { {\epsilon _ {2} }} \end{array} B \mathop \rightarrow \limits ^ \nu C$$

in $\mathfrak A$, where $( \epsilon _ {1} , \epsilon _ {2} )$ is the kernel pair of $\nu$, and $\nu$ is the cokernel of the pair $( \epsilon _ {1} , \epsilon _ {2} )$, into a diagram in $\mathfrak B$ with the same properties.