-- Module: Mathlib.RingTheory.Etale.StandardEtale
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
P : StandardEtalePair R
refine_5 :
  Eq (α :=
    Localization.Away P.g →ₐ[R]
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
    (AlgHom.comp (B :=
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
        (AlgHom.comp
          (P.lift
            ((algebraMap R[X]
                  (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                    (Ideal.span
                      (singleton (α := Localization.Away P.g)
                        ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                          Localization.Away P.g)))) :
                (a : R[X]) →
                  Localization.Away P.g ⧸
                    Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
              X)
            ⋯)
          (Ideal.Quotient.liftₐ
            (Ideal.span
              (singleton (α := Localization.Away P.g)
                ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                  Localization.Away P.g)))
            (IsLocalization.liftAlgHom (R := R[X]) (M := Submonoid.powers P.g) (f := aeval P.X) ⋯) ⋯))
        (Ideal.Quotient.mkₐ R
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))) :
      Localization.Away P.g →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
    (AlgHom.comp
        (AlgHom.id R
          (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
            (Ideal.span
              (singleton (α := Localization.Away P.g)
                ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                  Localization.Away P.g)))))
        (Ideal.Quotient.mkₐ R
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))) :
      Localization.Away P.g →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
⊢ Eq (α :=
    HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))) →ₐ[R]
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
    (AlgHom.comp
        (P.lift
          ((algebraMap R[X]
                (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                  (Ideal.span
                    (singleton (α := Localization.Away P.g)
                      ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                        Localization.Away P.g)))) :
              (a : R[X]) →
                Localization.Away P.g ⧸
                  Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
            X)
          ⋯)
        (Ideal.Quotient.liftₐ
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))
          (IsLocalization.liftAlgHom (R := R[X]) (M := Submonoid.powers P.g) (f := aeval P.X) ⋯) ⋯) :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))) →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
    (AlgHom.id R
        (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))) :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))) →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))

Tactic:
apply Ideal.Quotient.algHom_ext

Premises used: Ideal.Quotient.algHom_ext