{M : Type u_1} →
  [inst : Monoid M] →
    {N : Type u_2} →
      [inst_1 : Monoid N] →
        {φ : M →* N} →
          {A : Type u_4} →
            [inst_2 : Monoid A] →
              [inst_3 : MulDistribMulAction M A] →
                {B : Type u_5} →
                  [inst_4 : Monoid B] →
                    [inst_5 : MulDistribMulAction N B] →
                      (toMulActionHom : A →ₑ[⇑φ] B) →
                        toMulActionHom.toFun 1 = 1 →
                          (∀ (x y : A),
                              toMulActionHom.toFun (x * y) = toMulActionHom.toFun x * toMulActionHom.toFun y) →
                            A →ₑ*[φ] B