-- 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 S n
f :
  { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
    f.comp (universalFactorizationMap R n m k hn) =
      (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S) p }
i : Fin k
⊢ Eq.{u_2 + 1} (α := S)
    ((AlgHom.comp (B := MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
            (AlgHom.restrictScalars R (S := R)
              (↑((fun
                        (q :
                          { q : MonicDegreeEq S m × MonicDegreeEq S k //
                            (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }) ↦
                      (Subtype.mk (p := fun (f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S) ↦
                          f.comp (universalFactorizationMap R n m k hn) =
                            (Equiv.symm (mapEquivMonic R S n) :
                                (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                              p)
                          (Algebra.TensorProduct.lift
                            ((Equiv.symm (mapEquivMonic R S m) :
                                (a : MonicDegreeEq S m) → MvPolynomial (Fin m) R →ₐ[R] S)
                              (↑q : MonicDegreeEq S m × MonicDegreeEq S k).1)
                            ((Equiv.symm (mapEquivMonic R S k) :
                                (a : MonicDegreeEq S k) → MvPolynomial (Fin k) R →ₐ[R] S)
                              (↑q : MonicDegreeEq S m × MonicDegreeEq S k).2)
                            ⋯)
                          ⋯ :
                        { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                          f.comp (universalFactorizationMap R n m k hn) =
                            (Equiv.symm (mapEquivMonic R S n) :
                                (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                              p }))
                    ((fun
                          (f :
                            { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                              f.comp (universalFactorizationMap R n m k hn) =
                                (Equiv.symm (mapEquivMonic R S n) :
                                    (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                                  p }) ↦
                        (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                            (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]))
                            ((mapEquivMonic R S m : (a : MvPolynomial (Fin m) R →ₐ[R] S) → MonicDegreeEq S m)
                                ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                                  Algebra.TensorProduct.includeLeft),
                              (mapEquivMonic R S k : (a : MvPolynomial (Fin k) R →ₐ[R] S) → MonicDegreeEq S k)
                                ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                                  Algebra.TensorProduct.includeRight))
                            ⋯ :
                          { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }))
                      f)) :
                MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S))
            Algebra.TensorProduct.includeRight :
          (a : MvPolynomial (Fin k) R) → S)
        (X i) :
      S)
    ((AlgHom.comp (AlgHom.restrictScalars R (↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S))
            Algebra.TensorProduct.includeRight :
          (a : MvPolynomial (Fin k) R) → S)
        (X i) :
      S)

Tactic:
simp