-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Limits.HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
hf :
  IsOpenMap (X := (↑X : Type v)) (Y := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
ℱ 𝒢 : TopCat.Presheaf C Y
u : ℱ ⟶ 𝒢
U : Opens (↑X : Type v)
⊢ Eq (α := (obj (pullback C f) ℱ).obj (op U) ⟶ 𝒢.obj (op (hf.functor.obj U)))
    (NatTrans.app (CategoryTheory.Functor.map (pullback C f) u) (op U) ≫
      Iso.hom
        (Limits.IsColimit.coconePointUniqueUpToIso
          ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit 𝒢 (op U))
          (Limits.colimitOfDiagramTerminal
            {
              lift :=
                fun
                  (s :
                    Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                      (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
              fac := ⋯, uniq := ⋯ }
            (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ 𝒢))))
    (CategoryStruct.comp (Y := ℱ.obj (op (hf.functor.obj U)))
      (Iso.hom
          (Limits.IsColimit.coconePointUniqueUpToIso
            ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
            (Limits.colimitOfDiagramTerminal
              {
                lift :=
                  fun
                    (s :
                      Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                        (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                  CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
                fac := ⋯, uniq := ⋯ }
              (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))) :
        (obj (pullback C f) ℱ).obj (op U) ⟶ ℱ.obj (op (hf.functor.obj U)))
      (u.app (op (hf.functor.obj U)) : ℱ.obj (op (hf.functor.obj U)) ⟶ 𝒢.obj (op (hf.functor.obj U))))
have w : ∀ (j : CostructuredArrow (Opens.map f).op (op U)),
  Eq (α := obj (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ) j ⟶ 𝒢.obj (op (hf.functor.obj U)))
    (CategoryStruct.comp (Y :=
      obj
        (obj (const (CostructuredArrow (Opens.map f).op (op U)))
          (Limits.Cocone.pt (J := CostructuredArrow (Opens.map f).op (op U))
            ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op U))))
        j)
      (NatTrans.app
        (Limits.Cocone.ι
          ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
            (op U)))
        j)
      (NatTrans.app (CategoryTheory.Functor.map (pullback C f) u) (op U) ≫
        Iso.hom
          (Limits.IsColimit.coconePointUniqueUpToIso
            ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit 𝒢 (op U))
            (Limits.colimitOfDiagramTerminal
              {
                lift :=
                  fun
                    (s :
                      Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                        (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                  CostructuredArrow.homMk (homOfLE sorry).op sorry,
                fac := sorry, uniq := sorry }
              (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ 𝒢)))))
    (CategoryStruct.comp (Y :=
      obj
        (obj (const (CostructuredArrow (Opens.map f).op (op U)))
          (Limits.Cocone.pt (J := CostructuredArrow (Opens.map f).op (op U))
            ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op U))))
        j)
      (NatTrans.app
        (Limits.Cocone.ι
          ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
            (op U)))
        j)
      (CategoryStruct.comp (Y := ℱ.obj (op (hf.functor.obj U)))
        (Iso.hom
            (Limits.IsColimit.coconePointUniqueUpToIso
              ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
              (Limits.colimitOfDiagramTerminal
                {
                  lift :=
                    fun
                      (s :
                        Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                          (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                    CostructuredArrow.homMk (homOfLE sorry).op sorry,
                  fac := sorry, uniq := sorry }
                (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))) :
          (obj (pullback C f) ℱ).obj (op U) ⟶ ℱ.obj (op (hf.functor.obj U)))
        (u.app (op (hf.functor.obj U)) : ℱ.obj (op (hf.functor.obj U)) ⟶ 𝒢.obj (op (hf.functor.obj U)))))
Tactic: refine ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U)).hom_ext
    (fun j ↦ ?_)