-- 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
i : Fin m
refine_1 :
  Eq.{u_2 + 1} (α := S)
    ((f : (a : 𝓡) → S)
        ((fromTensor m k hn p : (a : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) → 𝓡)
            (MvPolynomial.X i ⊗ₜ[R] 1) :
          𝓡) :
      S)
    ((f : (a : R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)) → S)
        (1 ⊗ₜ[MvPolynomial (Fin n) R] (MvPolynomial.X i ⊗ₜ[R] 1)) :
      S)
⊢ Eq.{u_2 + 1} (α := S)
    ((AlgHom.comp (B := MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
            (AlgHom.restrictScalars R (S := MvPolynomial (Fin n) R)
              (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))
            Algebra.TensorProduct.includeLeft :
          (a : MvPolynomial (Fin m) R) → S)
        (MvPolynomial.X i) :
      S)
    ((AlgHom.comp
            (AlgHom.restrictScalars R
              (AlgHom.comp (AlgHom.restrictScalars (MvPolynomial (Fin n) R) f) Algebra.TensorProduct.includeRight))
            Algebra.TensorProduct.includeLeft :
          (a : MvPolynomial (Fin m) R) → S)
        (MvPolynomial.X i) :
      S)

Tactic:
simp [MvPolynomial.universalFactorizationMapLiftEquiv, MvPolynomial.mapEquivMonic,
        UniversalFactorizationRing.factor₁, coeff_freeMonic]

Premises used: MvPolynomial.universalFactorizationMapLiftEquiv, Polynomial.UniversalFactorizationRing.factor₁, MvPolynomial.mapEquivMonic, Polynomial.coeff_freeMonic