-- 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
apply :
  Eq (α := 𝓡 →ₐ[R] S)
    (AlgHom.comp (B := 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]) ∧
                  IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }) ↦
          (IsLocalization.liftAlgHom (R := 𝓡) (M :=
              Submonoid.powers (UniversalFactorizationRing.presentation m k hn p).jacobian) (f :=
              (Equiv.symm (UniversalFactorizationRing.homEquiv S m k hn p) :
                  (a :
                      { q : MonicDegreeEq S m × MonicDegreeEq S k //
                        (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) →
                    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]))
                  (↑q : MonicDegreeEq S m × MonicDegreeEq S 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]) ∧
                  IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]))
                (↑((UniversalFactorizationRing.homEquiv S m k hn p :
                        (a : 𝓡 →ₐ[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.comp (IsScalarTower.toAlgHom R 𝓡 𝓡'))) :
                  MonicDegreeEq S m × MonicDegreeEq S k)
                ⋯ :
              { q : MonicDegreeEq S m × MonicDegreeEq S k //
                (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧
                  IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }))
          f))
      (Algebra.algHom R 𝓡 𝓡'))
    (f.comp (Algebra.algHom R 𝓡 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]) ∧
                IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }) ↦
        (IsLocalization.liftAlgHom (R := 𝓡) (M :=
            Submonoid.powers (UniversalFactorizationRing.presentation m k hn p).jacobian) (f :=
            (Equiv.symm (UniversalFactorizationRing.homEquiv S m k hn p) :
                (a :
                    { q : MonicDegreeEq S m × MonicDegreeEq S k //
                      (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) →
                  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]))
                (↑q : MonicDegreeEq S m × MonicDegreeEq S 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]) ∧ IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]))
              (↑((UniversalFactorizationRing.homEquiv S m k hn p :
                      (a : 𝓡 →ₐ[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.comp (IsScalarTower.toAlgHom R 𝓡 𝓡'))) :
                MonicDegreeEq S m × MonicDegreeEq S k)
              ⋯ :
            { q : MonicDegreeEq S m × MonicDegreeEq S k //
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧
                IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }))
        f) =
    f

Tactic:
apply IsLocalization.algHom_ext
      (.powers (UniversalFactorizationRing.presentation m k hn p).jacobian)

Premises used: IsLocalization.algHom_ext, Algebra.PreSubmersivePresentation.jacobian, Polynomial.UniversalFactorizationRing.presentation, Submonoid.powers