-- Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
n m k : ℕ
hn : n = m + k
i : Fin n
hX :
  Eq (α := MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    ((aeval fun (x : Fin n) ↦
            (HMul.hMul (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
                  (Polynomial.map
                    (↑Algebra.TensorProduct.includeLeft :
                      MvPolynomial (Fin m) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
                    (freeMonic S m))
                  (Polynomial.map
                    (↑Algebra.TensorProduct.includeRight :
                      MvPolynomial (Fin k) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
                    (freeMonic S k))).coeff
              (↑x : ℕ) :
          (a : MvPolynomial (Fin n) S) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (((map (algebraMap R S) : RingHom.{u_1, u_2} (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) S)) :
            (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin n) S)
          (X i)) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    (((Algebra.TensorProduct.lift (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
              (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight) (mapAlgHom (Algebra.ofId R S)))
              ⋯ :
            AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
              (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
          (a : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        ((aeval fun (x : Fin n) ↦
              (HMul.hMul (α := (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)[X])
                    (Polynomial.map
                      (↑Algebra.TensorProduct.includeLeft :
                        MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (freeMonic R m))
                    (Polynomial.map
                      (↑Algebra.TensorProduct.includeRight :
                        MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (freeMonic R k))).coeff
                (↑x : ℕ) :
            (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
          (X i)) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
⊢ Eq.{u_2 + 1} (α := MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    (((RingHom.comp (AlgHom.toRingHom (universalFactorizationMap S n m k hn)) (map (algebraMap R S)) :
            RingHom.{u_1, u_2} (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
          (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (X i) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    (((RingHom.comp
              (AlgHom.toRingHom
                (Algebra.TensorProduct.lift
                    (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
                    (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                      (mapAlgHom (Algebra.ofId R S)))
                    ⋯ :
                  AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                    (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)))
              (AlgHom.toRingHom (universalFactorizationMap R n m k hn)) :
            RingHom.{u_1, u_2} (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
          (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (X i) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)

Tactic:
dsimp [universalFactorizationMap, mapEquivMonic]

Premises used: MvPolynomial.universalFactorizationMap, MvPolynomial.mapEquivMonic