-- Lean 4 proof step from Mathlib.CategoryTheory.SmallObject.WellOrderInductionData (goal)
Goal:
J : Type u
inst : LinearOrder J
inst_1 : SuccOrder J
F : Functor.{u, v, u, v + 1} Jᵒᵖ (Type v)
d : F.WellOrderInductionData
inst_2 : OrderBot J
val₀ : F.obj (op ⊥)
inst_3 : WellFoundedLT J
j : J
hj : Order.IsSuccLimit j
e : (i : J) → i < j → d.Extension val₀ i
i : J
hi : Order.IsSuccLimit i
hij : i ≤ j
hij' : i < j
this :
  (e i hij').val =
    d.lift i hi
      (Subtype.mk (p := fun (x : (j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))) ↦
        Membership.mem (γ := Set ((j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))))
          (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio i)) ⋯) ⋙ F)) x)
        (fun (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) ↦
          (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : F.obj (op i)) → F.obj (op (↑x.1 : J)))
            (e i hij').val)
        ⋯)
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (homOfLE hij).op) : (a : F.obj (op j)) → F.obj (op i))
      (d.lift j hj
        (Subtype.mk (p := fun (x : (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))) ↦
          Membership.mem (γ := Set ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))))
            (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio j)) ⋯) ⋙ F)) x)
          (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦ val (e (↑x.1 : J) ⋯)) ⋯)) =
    d.lift i hi
      (Subtype.mk (p := fun (x : (j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))) ↦
        Membership.mem (γ := Set ((j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))))
          (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio i)) ⋯) ⋙ F)) x)
        (fun (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) ↦
          (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : F.obj (op j)) → F.obj (op (↑x.1 : J)))
            (d.lift j hj
              (Subtype.mk (p := fun (x : (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))) ↦
                Membership.mem (γ := Set ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))))
                  (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio j)) ⋯) ⋙ F))
                  x)
                (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦ val (e (↑x.1 : J) ⋯)) ⋯)))
        ⋯)
have inl : (↑(Subtype.mk (p := fun (x : (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))) ↦
          Membership.mem (γ := Set ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))))
            (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio j)) sorry) ⋙ F)) x)
          (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦ val (e (↑x.1 : J) sorry)) sorry) :
      (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → (sorry.functor.op ⋙ F).obj j_1)
    (op (Subtype.mk (p := fun (x : J) ↦ x ∈ Set.Iio j) i hij')) =
  d.lift i hi
    (Subtype.mk (p := fun (x : (j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))) ↦
      Membership.mem (γ := Set ((j : { x : J // x ∈ Set.Iio i }ᵒᵖ) → F.obj (op (↑(unop j) : J))))
        (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio i)) sorry) ⋙ F)) x)
      (fun (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (homOfLE sorry))) :
            (a : F.obj (op j)) → F.obj (op (↑x.1 : J)))
          (d.lift j hj
            (Subtype.mk (p := fun (x : (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))) ↦
              Membership.mem (γ := Set ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) → F.obj (op (↑(unop j_1) : J))))
                (sections (Functor.op (Monotone.functor (f := Subtype.val (p := fun (x : J) ↦ x ∈ Set.Iio j)) sorry) ⋙ F))
                x)
              (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦ val (e (↑x.1 : J) sorry)) sorry)))
      sorry)
Tactic: rw [d.map_lift _ _ _ _ hij']