Given a ring R, denote by mod-R the category of nitely presented right R-modules, the two
functor categories (mod-R, Ab) and ((mod-R) op, Ab) with values in the category Ab of
abelian groups have received extensive attention since the s. Particularly they play
important roles in the investigation of the model theory of modules and the representation
theory of Artinian algebras (see eg [,,,,,,,,,,]). Strikingly, Herzog has developed a general
theory for the category (mod-R, Ab) and ((mod-R) op, Ab) for any ring R (see [,]). It is well …