In his thesis "Comparison of the Categories of Motives defined by Voevodsky and Nori (2016)Comparison of the Categories of Motives defined by Voevodsky and Nori (2016)" Daniel Harrer Compares "V. Voevodsky's geometric motives to the derived category of M. Nori's Abelian category of mixed motives by constructing a triangulated tensor functor between them."(compatible with the Betti realizations on both sides).