-- Module: Mathlib.Topology.Sheaves.Presheaf
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)
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 ⋯).op ⋯,
                  fac := ⋯, uniq := ⋯ }
                (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 ⋯).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)))))
⊢ 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))))

Tactic:
refine ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U)).hom_ext
    (fun j ↦ ?_)

Premises used: CategoryTheory.Functor.op, CategoryTheory.Limits.IsColimit.hom_ext, Opposite.op, TopologicalSpace.Opens.map, CategoryTheory.Functor.isPointwiseLeftKanExtensionLeftKanExtensionUnit