-- Lean 4 proof step from Mathlib.Algebra.Category.AlgCat.Limits (goal)
Goal:
R : Type u
inst : CommRing R
J : Type v
inst_1 : Category.{t, v} J
F : J ⥤ AlgCat.{w, u} R
inst_2 :
  Small.{w, max v w}
    (↑(F ⋙
            forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
              carrier)).sections :
      Type (max v w))
s : Cone F
r : R
⊢ Eq (α :=
    { x :
      (j : J) →
        (F ⋙
              forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                (CC := carrier)).obj
          j //
      x ∈
        (F ⋙
            forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
              carrier)).sections })
    (Subtype.mk (p :=
      fun
        (x :
          (j : J) →
            (F ⋙
                  forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                    (CC := carrier)).obj
              j) ↦
      x ∈
        (F ⋙
            forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
              carrier)).sections)
      (fun (j : J) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cone.π
                  (Functor.mapCone
                    (forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦
                      (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC := carrier))
                    s))
                j) :
            (a : ((Functor.const J).obj ((forget (AlgCat R)).mapCone s).pt).obj j) → (F ⋙ forget (AlgCat R)).obj j)
          ((algebraMap R (↑s.1 : Type w) : (a : R) → (↑s.1 : Type w)) r : (↑s.1 : Type w)))
      ⋯)
    ((algebraMap R ↥(sectionsSubalgebra F) : (a : R) → ↥(sectionsSubalgebra F)) r)
have refine_5 : Eq (α :=
  (j : J) →
    (F ⋙
          forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
            carrier)).obj
      j)
  (↑(Subtype.mk (p :=
        fun
          (x :
            (j : J) →
              (F ⋙
                    forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                      (CC := carrier)).obj
                j) ↦
        x ∈
          (F ⋙
              forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                (CC := carrier)).sections)
        (fun (j : J) ↦
          (ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (Cone.π
                    (Functor.mapCone
                      (forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦
                        (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC := carrier))
                      s))
                  j) :
              (a : ((Functor.const J).obj ((forget (AlgCat R)).mapCone s).pt).obj j) → (F ⋙ forget (AlgCat R)).obj j)
            ((algebraMap R (↑s.1 : Type w) : (a : R) → (↑s.1 : Type w)) r : (↑s.1 : Type w)))
        sorry) :
    (j : J) → (F ⋙ forget (AlgCat R)).obj j)
  (↑((algebraMap R ↥(sectionsSubalgebra F) : (a : R) → ↥(sectionsSubalgebra F)) r) :
    (j : J) → (F ⋙ forget (AlgCat R)).obj j)
Tactic: apply Subtype.ext