Let $ C $ be the category of exact categories (choose a Grothendieck universe), whose morphisms are adjoined pairs. For a category $ C $, let $ text {Ab} (C) $ be the category of objects of the abelian group $ C $. $ text {Ab} (C) $ is an abelian category.

East $ text {Ab} (C) $ the universal abelian category with a morphism $ p: C rightarrow text {Ab} (C) $ in $ C $?