The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those statements that involve "infinite constructions".

So I wonder: for which Grothendieck abelian $A$ (this certainly implies that $A$ is not small if it is non-zero) there exists an exact conservative functor $F$ into abelian groups (this is a certain weak substitute of "having elements")? Does the Gabriel-Popescu theorem help here (so, what can one say if $A$ is described as a "nice" localization of certain category of modules)?

Also, how would you call a functor $F$ possessing this property; does "a stalk functor" sound fine? What is the relation of the existence of $F$ condition to the existence of compact generators for $D(A)$? If $A$ is a category of sheaves for certain Grothendieck topology then can one relate $F$ to the points of this topology?

Any hints, references or examples are very welcome!