-- 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 :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Quiver.Hom.op (homOfLE ⋯))) :
        (a : (fun (X : Type v) ↦ X) (F.obj (op i))) → (fun (X : Type v) ↦ X) (F.obj (op i)))
      (e i hij').val =
    d.lift i hi
      (Subtype.mk (p :=
        fun
          (x :
            (j : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                j) ↦
        Membership.mem (γ :=
          Set
            ((j : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                j))
          (sections
            (Functor.op
                (Monotone.functor (f :=
                  (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
              F))
          x)
        (fun (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) ↦
          match (motive :=
            (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                x)
            x with
          | op ⟨k, hk⟩ =>
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (homOfLE ⋯).op) :
                (a : (fun (X : Type v) ↦ X) (F.obj (op i))) → (fun (X : Type v) ↦ X) (F.obj (op k)))
              (e i hij').val)
        ⋯)
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (homOfLE hij).op) :
        (a : (fun (X : Type v) ↦ X) (F.obj (op j))) → (fun (X : Type v) ↦ X) (F.obj (op i)))
      (d.lift j hj
        (Subtype.mk (p :=
          fun
            (x :
              (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                obj
                  (Functor.op
                      (Monotone.functor (f :=
                        (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) : (a : { x : J // x ∈ Set.Iio j }) → J))
                        ⋯) ⋙
                    F)
                  j_1) ↦
          Membership.mem (γ :=
            Set
              ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                obj
                  (Functor.op
                      (Monotone.functor (f :=
                        (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) : (a : { x : J // x ∈ Set.Iio j }) → J))
                        ⋯) ⋙
                    F)
                  j_1))
            (sections
              (Functor.op
                  (Monotone.functor (f :=
                    (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) : (a : { x : J // x ∈ Set.Iio j }) → J)) ⋯) ⋙
                F))
            x)
          (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦
            match (motive :=
              (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                obj
                  (Functor.op
                      (Monotone.functor (f :=
                        (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) : (a : { x : J // x ∈ Set.Iio j }) → J))
                        ⋯) ⋙
                    F)
                  x)
              x with
            | op ⟨i, hi⟩ => (e i hi).val)
          ⋯)) =
    d.lift i hi
      (Subtype.mk (p :=
        fun
          (x :
            (j : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                j) ↦
        Membership.mem (γ :=
          Set
            ((j : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                j))
          (sections
            (Functor.op
                (Monotone.functor (f :=
                  (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
              F))
          x)
        (fun (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) ↦
          match (motive :=
            (x : { x : J // x ∈ Set.Iio i }ᵒᵖ) →
              obj
                (Functor.op
                    (Monotone.functor (f :=
                      (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio i) : (a : { x : J // x ∈ Set.Iio i }) → J)) ⋯) ⋙
                  F)
                x)
            x with
          | op ⟨k, hk⟩ =>
            (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (homOfLE ⋯).op) :
                (a : (fun (X : Type v) ↦ X) (F.obj (op j))) → (fun (X : Type v) ↦ X) (F.obj (op k)))
              (d.lift j hj
                (Subtype.mk (p :=
                  fun
                    (x :
                      (j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                        obj
                          (Functor.op
                              (Monotone.functor (f :=
                                (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) :
                                  (a : { x : J // x ∈ Set.Iio j }) → J))
                                ⋯) ⋙
                            F)
                          j_1) ↦
                  Membership.mem (γ :=
                    Set
                      ((j_1 : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                        obj
                          (Functor.op
                              (Monotone.functor (f :=
                                (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) :
                                  (a : { x : J // x ∈ Set.Iio j }) → J))
                                ⋯) ⋙
                            F)
                          j_1))
                    (sections
                      (Functor.op
                          (Monotone.functor (f :=
                            (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) :
                              (a : { x : J // x ∈ Set.Iio j }) → J))
                            ⋯) ⋙
                        F))
                    x)
                  (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦
                    match (motive :=
                      (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) →
                        obj
                          (Functor.op
                              (Monotone.functor (f :=
                                (⇑(OrderHom.Subtype.val fun (x : J) ↦ x ∈ Set.Iio j) :
                                  (a : { x : J // x ∈ Set.Iio j }) → J))
                                ⋯) ⋙
                            F)
                          x)
                      x with
                    | op ⟨i, hi⟩ => (e i hi).val)
                  ⋯)))
        ⋯)
have inl : (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)) sorry) ⋙ F)) x)
        (fun (x : { x : J // x ∈ Set.Iio j }ᵒᵖ) ↦ val (e (↑x.1 : J) sorry)) sorry)) =
  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: dsimp at this ⊢