-- 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
⊢ EquivLike.coe (E :=
      (↑(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 w v)) ≃
        Shrink.{w, max w v}
          (↑(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 w v)))
      (equivShrink
        (↑(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 w v)))
      (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 : (fun (X : Type w) ↦ X) (((Functor.const J).obj ((forget (AlgCat R)).mapCone s).pt).obj j)) →
                (fun (X : Type w) ↦ X) ((F ⋙ forget (AlgCat R)).obj j))
            0)
        ⋯) =
    0
have refine_3 : ∀ (j : J),
  Eq.{w + 1} (α :=
    (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)
    ((↑((Equiv.symm
                (equivShrink
                  (↑(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 w v))) :
              (a : Shrink.{w, max w v} (↑(F ⋙ forget (AlgCat R)).sections : Type (max w v))) →
                (↑(F ⋙ forget (AlgCat R)).sections : Type (max w v)))
            (EquivLike.coe (E :=
              (↑(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 w v)) ≃
                Shrink.{w, max w v}
                  (↑(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 w v)))
              (equivShrink
                (↑(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 w v)))
              (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 : (fun (X : Type w) ↦ X) (((Functor.const J).obj ((forget (AlgCat R)).mapCone s).pt).obj j)) →
                        (fun (X : Type w) ↦ X) ((F ⋙ forget (AlgCat R)).obj j))
                    0)
                sorry))) :
        (j : J) → (F ⋙ forget (AlgCat R)).obj j)
      j)
    ((↑((Equiv.symm
                (equivShrink
                  (↑(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 w v))) :
              (a : Shrink.{w, max w v} (↑(F ⋙ forget (AlgCat R)).sections : Type (max w v))) →
                (↑(F ⋙ forget (AlgCat R)).sections : Type (max w v)))
            0) :
        (j : J) → (F ⋙ forget (AlgCat R)).obj j)
      j)
Tactic: ext j