-- Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Sheaf.{w, v_1, u_1} C X
U V : Opens (↑X : Type w)
s_1 :
  PullbackCone (X := F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).obj (op U))
    (Y := F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).obj (op V))
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
      (Quiver.Hom.op (homOfLE ⋯)))
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
      (Quiver.Hom.op (homOfLE ⋯)))
ι : ULift.{w, 0} WalkingPair → Opens (↑X : Type w) :=
  fun (x : ULift.{w, 0} WalkingPair) ↦
    match (motive := ULift.{w, 0} WalkingPair → Opens (↑X : Type w)) x with
    | { down := j } => WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦ Opens (↑X : Type w)) j U V
hι : U ⊔ V = iSup ι
s :
  PullbackCone (X := F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).obj (op U))
    (Y := F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).obj (op V))
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
      (Quiver.Hom.op (homOfLE ⋯)))
    (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
      (Quiver.Hom.op (homOfLE ⋯)))
m : s.pt ⟶ (F.interUnionPullbackCone U V).pt
h₁ : m ≫ (F.interUnionPullbackCone U V).fst = s.fst
h₂ : m ≫ (F.interUnionPullbackCone U V).snd = s.snd
a : ULift.{w, 0} WalkingPair
property :
  Eq (α :=
    s.pt ⟶
      Functor.obj
        ({
            obj := fun (X_1 : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) ↦
              op (Functor.obj (Pairwise.diagram ι) (unop X_1)),
            map := fun {X_1 Y : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ} (f : X_1 ⟶ Y) ↦
              Quiver.Hom.op (CategoryTheory.Functor.map (Pairwise.diagram ι) f.unop),
            map_id := ⋯, map_comp := ⋯ } ⋙
          F.presheaf)
        (op (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)))
    (m ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (eqToHom ⋯) ≫
            CategoryStruct.comp (Y := op (ι { down := WalkingPair.left }))
              (Quiver.Hom.op (homOfLE ⋯) :
                op
                    (Functor.obj
                      (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)))
                        (Cocone.pt (Pairwise.cocone ι)))
                      (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)) ⟶
                  op (ι { down := WalkingPair.left }))
              (Quiver.Hom.op (homOfLE ⋯) :
                op (ι { down := WalkingPair.left : ULift.{w, 0} WalkingPair }) ⟶
                  op
                    (Functor.obj (Pairwise.diagram ι)
                      (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)))) :
      s.pt ⟶
        Functor.obj
          ({
              obj := fun (X_1 : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) ↦
                op (Functor.obj (Pairwise.diagram ι) (unop X_1)),
              map := fun {X_1 Y : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ} (f : X_1 ⟶ Y) ↦
                Quiver.Hom.op (CategoryTheory.Functor.map (Pairwise.diagram ι) f.unop),
              map_id := ⋯, map_comp := ⋯ } ⋙
            F.presheaf)
          (op (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)))
    (F.interUnionPullbackConeLift U V s ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (eqToHom ⋯) ≫
            CategoryStruct.comp (Y := op (ι { down := WalkingPair.left }))
              (Quiver.Hom.op (homOfLE ⋯) :
                op
                    (Functor.obj
                      (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)))
                        (Cocone.pt (Pairwise.cocone ι)))
                      (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)) ⟶
                  op (ι { down := WalkingPair.left }))
              (Quiver.Hom.op (homOfLE ⋯) :
                op (ι { down := WalkingPair.left : ULift.{w, 0} WalkingPair }) ⟶
                  op
                    (Functor.obj (Pairwise.diagram ι)
                      (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)))) :
      s.pt ⟶
        Functor.obj
          ({
              obj := fun (X_1 : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) ↦
                op (Functor.obj (Pairwise.diagram ι) (unop X_1)),
              map := fun {X_1 Y : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ} (f : X_1 ⟶ Y) ↦
                Quiver.Hom.op (CategoryTheory.Functor.map (Pairwise.diagram ι) f.unop),
              map_id := ⋯, map_comp := ⋯ } ⋙
            F.presheaf)
          (op (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair } a)))
⊢ Eq (α :=
    s.pt ⟶ (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.pair { down := WalkingPair.left } a)))
    (m ≫
      F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
        (Quiver.Hom.op (eqToHom ⋯) ≫
          NatTrans.app (Cone.π (Cocone.op (Pairwise.cocone ι))) (op (Pairwise.pair { down := WalkingPair.left } a))))
    (F.interUnionPullbackConeLift U V s ≫
      F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
        (Quiver.Hom.op (eqToHom ⋯) ≫
          NatTrans.app (Cone.π (Cocone.op (Pairwise.cocone ι))) (op (Pairwise.pair { down := WalkingPair.left } a))))

Tactic:
dsimp only [Functor.op, Pairwise.cocone_ι_app, Functor.mapCone_π_app, Cocone.op,
        Pairwise.coconeιApp, unop_op, op_comp, NatTrans.op]

Premises used: CategoryTheory.Functor.op, CategoryTheory.op_comp, CategoryTheory.Pairwise.cocone_ι_app, CategoryTheory.NatTrans.op, CategoryTheory.Functor.mapCone_π_app, Opposite.unop_op, CategoryTheory.Pairwise.coconeιApp, CategoryTheory.Limits.Cocone.op