∀ {R : Type u_1} {A : Type u_2} {C : Type u_3} [inst : CommSemiring R] [inst_1 : NonUnitalSemiring A]
  [inst_2 : Module R A] [inst_3 : SMulCommClass R A A] [inst_4 : IsScalarTower R A A] [inst_5 : Semiring C]
  [inst_6 : Algebra R C] (f : A →ₙₐ[R] C), (Unitization.lift f).range = Algebra.adjoin R ↑(NonUnitalAlgHom.range f)