-- 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)
m :
  S.pt ⟶
    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)
hm :
  ∀
    (j :
      (ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
          ∃ (i : α), V ≤ Us i)ᵒᵖ),
    m ≫ NatTrans.app (Cone.π (mapCone (principalsKanExtension F) (opensLeCoverCocone Us).op)) j = S.π.app j
left : Discrete PUnit.{1}
x : (↑X : Type v)
f : obj (fromPUnit (Opposite.op (iSup Us))) left ⟶ obj (principals (↑X : Type v)) x
⊢ CategoryStruct.comp (Y :=
      limit (J := StructuredArrow (Opposite.op (iSup Us)) (principals (↑X : Type v))) (generator (iSup Us) ⋙ F))
      (limit.lift (generator (iSup Us) ⋙ F) (lowerCone Us S))
      (limit.π (generator (iSup Us) ⋙ F)
        { left := left, right := x, hom := f : Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }) =
    m ≫ limit.π (generator (iSup Us) ⋙ F) { left := left, right := x, hom := f }
have w : CategoryStruct.comp (Y :=
    limit (J :=
      StructuredArrow (D := (Opens (↑X : Type v))ᵒᵖ)
        (Opposite.op
          (obj (C :=
            ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
              ∃ (i : α), V ≤ Us i)
            (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)
            (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)
              (principalOpen
                (StructuredArrow.right
                  { left := left, right := x, hom := f :
                    Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }))
              sorry)))
        (principals (↑X : Type v)))
      (StructuredArrow.proj
          (Opposite.op
            (obj (C :=
              ObjectProperty.FullSubcategory (C := Opens (↑X : Type v)) fun (V : Opens (↑X : Type v)) ↦
                ∃ (i : α), V ≤ Us i)
              (ObjectProperty.ι fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)
              (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)
                (principalOpen
                  (StructuredArrow.right
                    { left := left, right := x, hom := f :
                      Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }))
                sorry)))
          (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)
          (principalOpen
            (StructuredArrow.right
              { left := left, right := x, hom := f :
                Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }))
          sorry)))
    (limit.π
      (generator
          (principalOpen
            (StructuredArrow.right
              { left := left, right := x, hom := f :
                Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) })) ⋙
        F)
      { left := { as := PUnit.unit },
        right :=
          StructuredArrow.right
            { left := left, right := x, hom := f :
              Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) },
        hom :=
          𝟙
            (Opposite.op
              (principalOpen
                (StructuredArrow.right
                  { left := left, right := x, hom := f :
                    Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }))) }) =
  m ≫
    limit.π (generator (iSup Us) ⋙ F)
      { left := left, right := x, hom := f : Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }
Tactic: simp only [lowerCone_pt, comp_obj, limit.lift_π, lowerCone_π_app, const_obj_obj, projSup_obj,
      op_obj, pointwiseRightKanExtension_obj]