-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections (goal)
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
i j : CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair)
g : j ⟶ i
⊢ 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 i) ⟶
      (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op j))
    (CategoryTheory.Functor.map
        (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) g.op ≫
      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 j) 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 ⋯))))
    ((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 i) 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 g.op)
have refine_2 : ∀ (g : j ⟶ Pairwise.single { down := WalkingPair.left }),
  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.single { down := WalkingPair.left : ULift.{w, 0} WalkingPair })) ⟶
      (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).obj (op j))
    (CategoryTheory.Functor.map
        (Functor.obj (Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) s.pt) g.op ≫
      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 j) 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 sorry)))
              (s.snd ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE sorry))))
    ((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.single { down := WalkingPair.left : 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 sorry)))
              (s.snd ≫
                F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))).map
                  (Quiver.Hom.op (homOfLE sorry)))) ≫
      (Functor.op (Pairwise.diagram ι) ⋙ F.presheaf).map g.op)
Tactic: rcases i with (⟨⟨_ | _⟩⟩ | ⟨⟨_ | _⟩, ⟨_⟩⟩)
Cleared hypotheses: i, g