# ct.category theory – Is every additive, left exact functor isomorphic to a hom functor?

Let $$A$$ be an Artin algebra, $$text{mod},A$$ the category of finitely generated $$A$$-modules and $$text{Ab}$$ the category of abelian groups. Is every additive, covariant, left-exact functor $$F:text{mod},A rightarrow text{Ab}$$ (natural) isomorphic to $$text{Hom}(X,-)$$ for some $$Xin text{mod},A$$? How do we obtain $$X$$?