  We define a cellular approximation for the diagonal of the Forcey--Loday
realizations of the multiplihedra, and endow them with a compatible topological
cellular operadic bimodule structure over the Loday realizations of the
associahedra. This provides us with a model for topological and algebraic
A-infinity morphisms, as well as a universal and explicit formula for their
tensor product. We study the monoidal properties of this newly defined tensor
product and conclude by outlining several applications, notably in algebraic
and symplectic topology.
