-- 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)
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
w : ∃ (i : α), principalOpen x ≤ Us i
w₁ :
  ∀
    (hm :
      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)
                (principalOpen x) ⋯)))
        (m ≫
          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)
                (principalOpen x) ⋯)))
        (S.π.app
          (Opposite.op
            (ObjectProperty.FullSubcategory.mk (P := fun (V : Opens (↑X : Type v)) ↦ ∃ (i : α), V ≤ Us i)
              (principalOpen 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
                    (StructuredArrow.right
                      { left := left, right := x, hom := f :
                        Comma (fromPUnit (Opposite.op (iSup Us))) (principals (↑X : Type v)) }))
                  ⋯)))
            (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)) }))
                    ⋯)))
              (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)) }))
              ⋯)))
        (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)) }
⊢ 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)) }))
                ⋯)))
          (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)) }))
                  ⋯)))
            (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)) }))
            ⋯)))
      (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:
specialize hm ⟨principalOpen x, ?_⟩

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.mk, Opposite.op, Alexandrov.principalOpen