On the Two Categories of Modules

A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules and their allowed morphisms, defined in the paper. The equivalence is expressed by using generators. In particular, we obtain the well-known equivalence of the two categories of projective finitely generated modules; thus, our main result extends this classical one.