-- 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
property :
  Eq (α :=
    s.pt ⟶ (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single { down := WalkingPair.left })))
    (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.single { down := WalkingPair.left }))))
    (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.single { down := WalkingPair.left }))))
⊢ Eq (α :=
    s.pt ⟶ (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single { down := WalkingPair.left })))
    ((m ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (eqToHom ⋯))) ≫
      (Functor.mapCone F.presheaf (Cocone.op (Pairwise.cocone ι))).π.app
        (op (Pairwise.single { down := WalkingPair.left })))
    ((F.interUnionPullbackConeLift U V s ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (eqToHom ⋯))) ≫
      (Functor.mapCone F.presheaf (Cocone.op (Pairwise.cocone ι))).π.app
        (op (Pairwise.single { down := WalkingPair.left })))

Tactic:
rw [Category.assoc, Category.assoc, Functor.mapCone_π_app, ← F.1.map_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.ObjectProperty.FullSubcategory.obj, CategoryTheory.Functor.mapCone_π_app, CategoryTheory.Functor.map_comp