Objects of which Grothendieck abelian categories have elements? 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!
 A: I am not an expert of the abelian world, but I think I can answer.
Since my background is not precisely abelian, I will start with an example in category theory.

Thm. Let $\mathcal{K}$ be a locally $\kappa$-presentable category, then there is a faithful and conservative functor to Set that preserves $\kappa$-directed colimits.

Proof. Since $\mathcal{K}$ is a locally $\kappa$-presentable category, it has a strong generator $\mathcal{G}$ made by $\kappa$-presentable objects. Then, the functor $\coprod_{G \in \mathcal{G}} \text{hom}_{\mathcal{K}}(G, \_)$ is faithful and conservative. Moreover it preserves $\kappa$-directed colimits and connected limits.
It looks to me that with the same argument one can prove the following statement.

Thm. Let $\mathcal{A}$ be an abelian category with a strong projective generator, then it has an exact, faithful and conservative functor into $\mathbb{Ab}$. Moreover, if the generator is made by $\kappa$-presentable objects, the functor preserves $\kappa$-directed colimits. 

Observe that the functor is left exact because $\mathbb{Ab}$ is AB4.
Moreover, the following holds.

Thm. Let $\mathcal{A}$ be a Grothendieck category with a faithful and conservative functor in $\mathbb{Ab}$ which is accessible, exact and preserve connected limits. Then $\mathcal{A}$ has a  projective strong generator.

Proof. Call $F$ such a functor, then it must have a multiadjoint. This is equivalent to the fact that $F \cong \coprod_{G \in \mathcal{G}} \text{hom}_{\mathcal{A}}(G, \_)$ for a small family $\mathcal{G}$. Since $F$ is faithful and conservative, $\mathcal{G}$ must be a strong generator.
