LetMbe ak-vector space andR∈ Endk(M⊗M)In [13] we introduced and studied what we called the Hopf equation:R12R23=R23R13R12.By means of a FRT type theorem, we have proven that the categoryofH-Hopf modules is deeply involved in solving this equation. In the present paper, we continue to study the Hopf equation from another perspective:having in mind the quantum Yang-Baxter equation, in the solution of which the co-quasitriangular (or braided) bialgebras play an important role (see [8]), we introduce and study what we call bialgebras with Hopf functions. The main theorem of this paper shows that, ifMis finite dimensional, any solutionRof the Hopf equation has the form R = Rσ. whereMis a right comodule over a bialgebra with a Hopf function (B{R),C,σ) andRσis the special map.