-- Lean 4 proof step from Mathlib.Topology.Sheaves.Alexandrov (goal)
Goal:
X_1 : Type v
inst : TopologicalSpace X_1
inst_1 : Preorder X_1
inst_2 : Topology.IsUpperSet X_1
C : Type u
inst_3 : Category.{v, u} C
inst_4 : HasLimits.{v, u} C
F_1 : X_1 ⥤ C
X : TopCat
inst_5 : Preorder (↑X : Type v)
inst_6 : Topology.IsUpperSet.{v} (↑X : Type v)
F : CategoryTheory.Functor.{v, v, v, u} (↑X : Type v) C
α : Type v
Us : α → Opens (↑X : Type v)
S :
  Cone (J :=
    (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)ᵒᵖ)
    (Functor.op (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) ⋙ principalsKanExtension F)
V : Opens (↑X : Type v)
i : α
hV : V ≤ Us i
⊢ Eq (α :=
    S.pt ⟶
      obj (C :=
        (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
            ∃ (i : α), V ≤ Us i)ᵒᵖ)
        (Functor.op (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) ⋙ principalsKanExtension F)
        (Opposite.op
          (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V ⋯)))
    (CategoryStruct.comp (Y :=
      Cone.pt (F :=
        Functor.op (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) ⋙ principalsKanExtension F)
        (mapCone (principalsKanExtension F) (opensLeCoverCocone Us).op))
      (limit.lift (generator (iSup Us) ⋙ F) (lowerCone Us S))
      (NatTrans.app (Cone.π (mapCone (principalsKanExtension F) (opensLeCoverCocone Us).op))
        (Opposite.op
          (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V ⋯))))
    (S.π.app
      (Opposite.op (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V ⋯)))
have dsimp : Eq (α :=
  S.pt ⟶
    limit (J := StructuredArrow (Opposite.op V) (principals (↑X : Type v)))
      (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F))
  (CategoryStruct.comp (Y :=
      limit (J := StructuredArrow (Opposite.op (iSup Us)) (principals (↑X : Type v)))
        (StructuredArrow.proj (Opposite.op (iSup Us)) (principals (↑X : Type v)) ⋙ F))
      (limit.lift (generator (iSup Us) ⋙ F) (lowerCone Us S))
      (limit.lift (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F)
        {
          pt :=
            limit (J := StructuredArrow (Opposite.op (iSup Us)) (principals (↑X : Type v)))
              (StructuredArrow.proj (Opposite.op (iSup Us)) (principals (↑X : Type v)) ⋙ F),
          π :=
            {
              app := fun (g : StructuredArrow (Opposite.op V) (principals (↑X : Type v))) ↦
                limit.π (StructuredArrow.proj (Opposite.op (iSup Us)) (principals (↑X : Type v)) ⋙ F)
                  (obj
                    (StructuredArrow.map
                      (CategoryStruct.comp (Y :=
                        Opposite.op
                          (Us
                            (OpensLeCover.index (U := Us)
                              (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦
                                ∃ (i : α), V ≤ Us i) V sorry))))
                        (Quiver.Hom.op
                          (Opens.leSupr Us
                            (OpensLeCover.index (U := Us)
                              (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦
                                ∃ (i : α), V ≤ Us i) V sorry))))
                        (Quiver.Hom.op
                          (OpensLeCover.homToIndex
                            (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦
                              ∃ (i : α), V ≤ Us i) V sorry)))))
                    g),
              naturality := sorry } }) :
    S.pt ⟶
      limit (J := StructuredArrow (Opposite.op V) (principals (↑X : Type v)))
        (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F))
  (S.π.app
      (Opposite.op (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V sorry)) :
    obj (C :=
        (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
            ∃ (i : α), V ≤ Us i)ᵒᵖ)
        (obj
          (Functor.const
            (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
                ∃ (i : α), V ≤ Us i)ᵒᵖ)
          S.pt)
        (Opposite.op
          (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V sorry)) ⟶
      obj (C :=
        (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
            ∃ (i : α), V ≤ Us i)ᵒᵖ)
        (Functor.op (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) ⋙ principalsKanExtension F)
        (Opposite.op
          (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i) V sorry)))
Tactic: dsimp [forget, opensLeCoverCocone]