Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and any object of $\mathcal{A}$ becomes projective in it (if we apply the Yoneda embedding for this setting). Now note that $PreSh(\mathcal{A})$ does not fully determine $\mathcal{A}$; so there can exist "natural" functors from $PreSh(\mathcal{A})$ into $PreSh(\mathcal{B})$ that do not come from additive functors from $B$ into $A$. Did anybody study functors of this sort?
Besides, any "formal" direct sum of $\mathcal{A}$-morphisms satisfying obvious finiteness conditions "acts" on objects of $PreSh(\mathcal{A})$. One can form a certain (associative unital) ring $R$ of formal direct sums; one can possibly describe $PreSh(\mathcal{A})$ as a certain modules over this ring. Now, I would like to localize this $R$ and to factorize it by certain (two-sided and non-homogeneous) ideals. So, I wonder whether one can "categorify" the corresponding full subcategories of $PreSh(\mathcal{A})$.
Any references or hints would be very welcome!