The binary product in the 2-category of triangular bialgebras and twisted morphisms

It is well-known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras and twisted morphisms of triangular bialgebras. We do so by adopting the framework of 2-categories and the proper notion of binary product, as well as by employing a description of twists on the tensor product bialgebra, specifically developed for this purpose. We apply this extension to provide a new interpretation of the twisted tensor products of triangular bialgebras in terms of binary products.