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!



Objects of which Grothendieck abelian categories have elements?
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.



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> **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.
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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. 
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Observe that the functor is left exact because $\mathbb{Ab}$ is AB4.


Moreover, the following holds.



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> **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.
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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.


