Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\mathrm{Hom}(A,-)$ for some $A\in Ab$?



Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?
The category of abelian groups is small-complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). It follows from the Special Adjoint Functor Theorem that any limit-preserving functor $G: Ab \to Ab$ has a left adjoint $F$. (A proof of the SAFT may be found on [this nLab page](https://ncatlab.org/nlab/show/adjoint+functor+theorem).) And as Dylan Wilson pointed out in a comment, we then have $G \cong \text{Hom}(\mathbb{Z}, G-) \cong \text{Hom}(F(\mathbb{Z}), -)$, so $G$ is representable. 


Referring to another comment by Dylan: all such functors are necessarily accessible, since any abelian group is $\kappa$-presentable for some $\kappa$. 


