A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into short exact sequences.
A covariant functor is called left exact if it preserves the exactness of all sequences
and it is called right exact if it preserves the exactness of all sequences
("Left" and "right" are interchanged in the corresponding definitions for contravariant functors.)
A functor is exact iff it is both left and right exact.
Every tensor product functor is right exact. For every module over a unit ring , the covariant functor and the contravariant functor are left exact; the first is exact iff is projective and the second iff is injective.