• left-exact if it transforms kernels into kernels
• right-exact if it transforms cokernels into cokernels
• exact if it is left exact and right exact, i.e. if it transforms exact sequences into exact sequences

• left-exact if it transforms cokernels into kernels
• right-exact it if transforms kernels into cokernels
• exact if it is left exact and right exact.

Examples

The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F_A(X) = Hom_A(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor F_A is exact if and only if A is projective. The functor G_A(X) = Hom_A(X,A) is a contravariant left-exact functor; it is exact if and only if A is injective.

If k is a field and V is a vector space over k, we write V* = Hom_k(V,k). This yields an exact functor from the category of k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.)

If X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The functor which associates to each sheave L the group of global sections L(X) is left-exact.

If R is a ring and T is a right R-module, we can define a functor H_T from the abelian category of all left R-modules to Ab by using the tensor product over R: H_T(X) = T ⊗ X. This is a covariant right exact functor; it is exact if and only if T is flat.

If A and B are two abelian categories, we can consider the functor category B^A consisting of all functors from A to B. If A is a given object of A, then we get a functor E_A from B^A to B by evaluating functors at A. This functor E_A is exact.

Some facts

Every equivalence or duality of abelian categories is exact.

A covariant (not necessarily additive) functor is left exact if and only if it turns finite limitss into limits; a covariant functor is right exact if and only if it turns finite colimits into colimits; a contravariant functor is left exact if and only if it turns finite colimits into limits; a covariant functor is right exact if and only if it turns finite limits into colimits.

The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functorss.