-- Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
f : 𝓡 →ₐ[R] S
this_1 : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=
  RingHom.toAlgebra (AlgHom.toRingHom (MvPolynomial.universalFactorizationMap R n m k hn))
this_2 : Algebra (MvPolynomial (Fin n) R) R :=
  RingHom.toAlgebra
    (AlgHom.toRingHom (R := R)
      ((Equiv.symm (MvPolynomial.mapEquivMonic R R n) : (a : MonicDegreeEq R n) → MvPolynomial (Fin n) R →ₐ[R] R) p))
this_3 : Algebra.{u_1, u_2} (MvPolynomial (Fin n) R) S :=
  Algebra.compHom (R := R) S
    (AlgHom.toRingHom (R := R)
      ((Equiv.symm (MvPolynomial.mapEquivMonic R R n) : (a : MonicDegreeEq R n) → MvPolynomial (Fin n) R →ₐ[R] R) p))
this_4 : IsScalarTower (MvPolynomial (Fin n) R) R S
this : IsScalarTower R (MvPolynomial (Fin n) R) S
refine :
  Eq (α := AlgHom.{u_1, u_1, u_2} (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) S)
    (AlgHom.comp (B := R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R))
      (AlgHom.restrictScalars (MvPolynomial (Fin n) R) (S := R)
        ((fun
              (q :
                { q : MonicDegreeEq S m × MonicDegreeEq S k //
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) ↦
            (Algebra.TensorProduct.lift (Algebra.ofId R S)
                (let __RingHom : RingHom.{u_1, u_2} (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) S :=
                  AlgHom.toRingHom (R := R)
                    (↑((Equiv.symm
                              (MvPolynomial.universalFactorizationMapLiftEquiv R S n m k hn (p.map (algebraMap R S))) :
                            (a :
                                { q : MonicDegreeEq S m × MonicDegreeEq S k //
                                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑(p.map (algebraMap R S)) : S[X]) }) →
                              { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                                f.comp (MvPolynomial.universalFactorizationMap R n m k hn) =
                                  ((MvPolynomial.mapEquivMonic R S n).symm :
                                      (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                                    (p.map (algebraMap R S)) })
                          q) :
                      MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S);
                { toRingHom := __RingHom, commutes' := ⋯ })
                ⋯ :
              R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →ₐ[R] S))
          ((fun (f : 𝓡 →ₐ[R] S) ↦
              (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]))
                  ((factor₁ m k hn p).map (↑f : 𝓡 →+* S), (factor₂ m k hn p).map (↑f : 𝓡 →+* S)) ⋯ :
                { q : MonicDegreeEq S m × MonicDegreeEq S k //
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }))
            f)))
      Algebra.TensorProduct.includeRight)
    (AlgHom.comp (AlgHom.restrictScalars (MvPolynomial (Fin n) R) f) Algebra.TensorProduct.includeRight)
⊢ (fun
          (q :
            { q : MonicDegreeEq S m × MonicDegreeEq S k //
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) ↦
        (Algebra.TensorProduct.lift (Algebra.ofId R S)
            (let __RingHom : RingHom.{u_1, u_2} (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) S :=
              AlgHom.toRingHom (R := R)
                (↑((Equiv.symm (MvPolynomial.universalFactorizationMapLiftEquiv R S n m k hn (p.map (algebraMap R S))) :
                        (a :
                            { q : MonicDegreeEq S m × MonicDegreeEq S k //
                              (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑(p.map (algebraMap R S)) : S[X]) }) →
                          { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                            f.comp (MvPolynomial.universalFactorizationMap R n m k hn) =
                              ((MvPolynomial.mapEquivMonic R S n).symm :
                                  (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                                (p.map (algebraMap R S)) })
                      q) :
                  MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S);
            { toRingHom := __RingHom, commutes' := ⋯ })
            ⋯ :
          R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →ₐ[R] S))
      ((fun (f : 𝓡 →ₐ[R] S) ↦
          (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]))
              ((factor₁ m k hn p).map (↑f : 𝓡 →+* S), (factor₂ m k hn p).map (↑f : 𝓡 →+* S)) ⋯ :
            { q : MonicDegreeEq S m × MonicDegreeEq S k //
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }))
        f) =
    f

Tactic:
refine Algebra.TensorProduct.ext (by ext) ?_

Premises used: Algebra.TensorProduct.ext