A categorical formulation of the Deligne-Terasoma approach to double shuffle theory

In this paper, we introduce the notion of a bimodule with a factorization structure (BFS) and show that such a structure gives rise to an algebra morphism. We then prove that this framework offers an interpretation of the geometric construction underlying both the Betti and de Rham harmonic coproducts of the double shuffle theory developed Enriquez-Furusho inspired by an unpublished preprint of Deligne-Terasoma.