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 $X$?