Recently I have heard somewhere that any (edit: small) abelian category can be expressed as the colimit of categories of projective modules over some rings. The remark was that this is "basically just idempotent completion". I have been trying to figure out myself how this works but got stuck. Maybe somebody knows how to do this or where I can find a good reference.

My idea so far was to consider the rings $R=End(A,A)$ for any object $A$ in the abelian category $\mathcal A$. Then I can consider the functor $h_A:\mathcal A\to R\text{-}mod$ which maps $X$ to the $R$ module $Hom(A,X)$, which gives by Yoneda an embedding $\mathcal A\to (R\text{-}mod)^\mathcal A$. $A$ itself and its coproducts are then precisely the free $R$-modules.

If I take the idempotent completion of the category of free $R$-modules, I get the category of projective $R$-modules.

But this all doesn't really seem to go anywhere. Am I on the right way?

Also this statement seems so strong that it is odd that I can't find a reference. If I have any funtorial property for projective $R$-modules which commutes with colimits (like so many things in K-theory) I automatically have it for any abelian category. Maybe this means that the statement should be weaker?