-- 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 :
  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 (j : ULift.{w, 0} WalkingPair) ↦
    WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦ Opens (↑X : Type w)) j.down U V
hι : U ⊔ V = iSup ι
down : WalkingPair
refine_2 :
  𝟙 s.pt ≫
      s.fst ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (homOfLE ⋯)) =
    (s.fst ≫
        F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
          (Quiver.Hom.op (homOfLE ⋯))) ≫
      F.presheaf.map
        (𝟙
          (op
            (ι { down := WalkingPair.left : ULift.{w, 0} WalkingPair } ⊓
              ι { down := down : ULift.{w, 0} WalkingPair })))
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{w, v_1, w, u_1}
        (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
        (op
          (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair }
            { down := down : ULift.{w, 0} WalkingPair })) ⟶
      (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj
        (op (Pairwise.pair { down := WalkingPair.left } { down := down })))
    (CategoryStruct.comp (Y :=
      Functor.obj.{w, v_1, w, u_1}
        (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
        (op
          (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair }
            { down := down : ULift.{w, 0} WalkingPair })))
      (CategoryTheory.Functor.map
        (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
        (Quiver.Hom.op (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down })))
      (Opposite.casesOn (motive := fun (x : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) ↦
        Functor.obj.{w, v_1, w, u_1}
            (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) x ⟶
          (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj x)
        (op
          (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair }
            { down := down : ULift.{w, 0} WalkingPair }))
        fun (unop : CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)) ↦
        Pairwise.casesOn (motive := fun (x : CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)) ↦
          Functor.obj.{w, v_1, w, u_1}
              (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) (op x) ⟶
            (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op x))
          unop
          (fun (a : ULift.{w, 0} WalkingPair) ↦
            ULift.casesOn (motive := fun (x : ULift.{w, 0} WalkingPair) ↦
              Functor.obj.{w, v_1, w, u_1}
                  (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                  (op (Pairwise.single x)) ⟶
                (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single x)))
              a fun (down : WalkingPair) ↦
              WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦
                Functor.obj.{w, v_1, w, u_1}
                    (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                    (op (Pairwise.single { down := x : ULift.{w, 0} WalkingPair })) ⟶
                  (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single { down := x })))
                down s.fst s.snd)
          fun (a a_1 : ULift.{w, 0} WalkingPair) ↦
          ULift.casesOn (motive := fun (x : ULift.{w, 0} WalkingPair) ↦
            Functor.obj.{w, v_1, w, u_1}
                (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                (op (Pairwise.pair x a_1)) ⟶
              (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.pair x a_1)))
            a fun (down : WalkingPair) ↦
            WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦
              Functor.obj.{w, v_1, w, u_1}
                  (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                  (op (Pairwise.pair { down := x : ULift.{w, 0} WalkingPair } a_1)) ⟶
                (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.pair { down := x } a_1)))
              down
              (s.fst ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE ⋯)))
              (s.snd ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE ⋯)))))
    (CategoryStruct.comp (Y :=
      (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj
        (op (Pairwise.pair { down := WalkingPair.left } { down := down })))
      (Opposite.casesOn (motive := fun (x : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) ↦
        Functor.obj.{w, v_1, w, u_1}
            (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) x ⟶
          (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj x)
        (op
          (Pairwise.pair { down := WalkingPair.left : ULift.{w, 0} WalkingPair }
            { down := down : ULift.{w, 0} WalkingPair }))
        fun (unop : CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)) ↦
        Pairwise.casesOn (motive := fun (x : CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)) ↦
          Functor.obj.{w, v_1, w, u_1}
              (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) (op x) ⟶
            (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op x))
          unop
          (fun (a : ULift.{w, 0} WalkingPair) ↦
            ULift.casesOn (motive := fun (x : ULift.{w, 0} WalkingPair) ↦
              Functor.obj.{w, v_1, w, u_1}
                  (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                  (op (Pairwise.single x)) ⟶
                (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single x)))
              a fun (down : WalkingPair) ↦
              WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦
                Functor.obj.{w, v_1, w, u_1}
                    (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                    (op (Pairwise.single { down := x : ULift.{w, 0} WalkingPair })) ⟶
                  (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.single { down := x })))
                down s.fst s.snd)
          fun (a a_1 : ULift.{w, 0} WalkingPair) ↦
          ULift.casesOn (motive := fun (x : ULift.{w, 0} WalkingPair) ↦
            Functor.obj.{w, v_1, w, u_1}
                (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                (op (Pairwise.pair x a_1)) ⟶
              (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.pair x a_1)))
            a fun (down : WalkingPair) ↦
            WalkingPair.casesOn (motive := fun (x : WalkingPair) ↦
              Functor.obj.{w, v_1, w, u_1}
                  (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt)
                  (op (Pairwise.pair { down := x : ULift.{w, 0} WalkingPair } a_1)) ⟶
                (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op (Pairwise.pair { down := x } a_1)))
              down
              (s.fst ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE ⋯)))
              (s.snd ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
      ((Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).map
        (Quiver.Hom.op (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down }))))

Tactic:
dsimp [Pairwise.diagram]

Premises used: CategoryTheory.Pairwise.diagram