∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
  [inst_3 : Bialgebra R A] [inst_4 : Bialgebra R B] (f : A ≃ₐ[R] B)
  (counit_comp : (Bialgebra.counitAlgHom R B).comp ↑f = Bialgebra.counitAlgHom R A)
  (map_comp_comul :
    (Algebra.TensorProduct.map ↑f ↑f).comp (Bialgebra.comulAlgHom R A) = (Bialgebra.comulAlgHom R B).comp ↑f)
  (a : A), (BialgEquiv.ofAlgEquiv f counit_comp map_comp_comul) a = f.toFun a