  We prove that a Hom-finite additive category having determined morphisms on
both sides is a dualizing variety. This complements a result by Krause. We
prove that in a Hom-finite abelian category having Serre duality, a morphism is
right determined by some object if and only if it is an epimorphism. We give a
characterization to abelian categories having Serre duality via determined
morphisms.

Representation Theory;Rings and Algebras