-- Module: Mathlib.Topology.Sheaves.Alexandrov
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
left : Discrete PUnit.{1}
x : (↑X : Type v)
f : obj (fromPUnit (Opposite.op V)) left ⟶ obj (principals (↑X : Type v)) x
w :
  Eq (α := S.pt ⟶ F.obj x)
    (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 x) ⋯)))
            (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 x) ⋯)))
              (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 x) ⋯)))
        (limit.π (generator (principalOpen x) ⋙ F)
          { left := { as := PUnit.unit }, right := x, hom := 𝟙 (Opposite.op (principalOpen x)) }) :
      S.pt ⟶ F.obj x)
    (CategoryStruct.comp (Y :=
        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 ⋯)) :
          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 ⋯)) ⟶
            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 ⋯)))
        (limit.π (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F)
          { left := left, right := x, hom := f }) :
      S.pt ⟶ F.obj x)
⊢ Eq (α :=
    S.pt ⟶
      obj (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F)
        { left := left, right := x, hom := f : Comma (fromPUnit (Opposite.op V)) (principals (↑X : Type v)) })
    (CategoryStruct.comp (Y :=
      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 ⋯))))
                          (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 ⋯))))
                          (Quiver.Hom.op
                            (OpensLeCover.homToIndex
                              (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦
                                ∃ (i : α), V ≤ Us i) V ⋯)))))
                      g),
                naturality := ⋯ } }))
      (limit.π (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F)
        { left := left, right := x, hom := f }))
    (CategoryStruct.comp (Y :=
      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 ⋯)) :
        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 ⋯)) ⟶
          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 ⋯)))
      (limit.π (StructuredArrow.proj (Opposite.op V) (principals (↑X : Type v)) ⋙ F)
        { left := left, right := x, hom := f }))

Tactic:
simp only [Category.assoc, limit.lift_π, lowerCone_pt, lowerCone_π_app, const_obj_obj,
      projSup_obj, StructuredArrow.map_obj_right, comp_obj, op_obj, pointwiseRightKanExtension_obj,
      StructuredArrow.proj_obj]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.StructuredArrow.map_obj_right, CategoryTheory.Category.assoc, Alexandrov.lowerCone_π_app, CategoryTheory.Functor.pointwiseRightKanExtension_obj, Alexandrov.projSup_obj, Alexandrov.lowerCone_pt, CategoryTheory.Functor.const_obj_obj, CategoryTheory.StructuredArrow.proj_obj, CategoryTheory.Limits.limit.lift_π, CategoryTheory.Functor.op_obj