-- Module: Mathlib.Topology.Sheaves.Presheaf
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U V : Opens (↑X : Type v)
HU :
  IsOpen (X := (↑Y : Type v))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f) :
        (a : (↑X : Type v)) → (↑Y : Type v)) ''
      (↑U : Set (↑X : Type v)))
HV :
  IsOpen (X := (↑Y : Type v))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f) :
        (a : (↑X : Type v)) → (↑Y : Type v)) ''
      (↑V : Set (↑X : Type v)))
le : U ≤ V
refine :
  ∀ (j : CostructuredArrow (Opens.map f).op (op V)),
    Eq (α :=
      obj (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ) j ⟶
        ℱ.obj
          (op
            {
              carrier :=
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                        f) :
                    (a : (↑X : Type v)) → (↑Y : Type v)) ''
                  (↑U : Set (↑X : Type v)),
              is_open' := HU }))
      (NatTrans.app
          (Cocone.ι
            ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)))
          j ≫
        (obj (pullback C f) ℱ).map (homOfLE le).op ≫
          Iso.hom
            (IsColimit.coconePointUniqueUpToIso
              ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
              (colimitOfDiagramTerminal
                {
                  lift :=
                    fun
                      (s :
                        Cone (C := CostructuredArrow (Opens.map f).op (op U))
                          (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                    CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
                  fac := ⋯, uniq := ⋯ :
                  IsLimit (C := CostructuredArrow (Opens.map f).op (op U))
                    (asEmptyCone
                      (CostructuredArrow.mk (Y :=
                        op
                          {
                            carrier :=
                              (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v)))
                                      (CC := carrier) f) :
                                  (a : (↑X : Type v)) → (↑Y : Type v)) ''
                                (↑U : Set (↑X : Type v)),
                            is_open' := HU : Opens (↑Y : Type v) })
                        (homOfLE ⋯).op)) }
                (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))))
      (CategoryStruct.comp (Y :=
        obj
          (obj (Functor.const (CostructuredArrow (Opens.map f).op (op V)))
            (Cocone.pt (J := CostructuredArrow (Opens.map f).op (op V))
              ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ)
                    ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
                (op V))))
          j)
        (NatTrans.app
          (Cocone.ι
            ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)))
          j)
        (Iso.hom
            (IsColimit.coconePointUniqueUpToIso
              ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op V))
              (colimitOfDiagramTerminal
                {
                  lift :=
                    fun
                      (s :
                        Cone (C := CostructuredArrow (Opens.map f).op (op V))
                          (empty (CostructuredArrow (Opens.map f).op (op V)))) ↦
                    CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
                  fac := ⋯, uniq := ⋯ }
                (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ))) ≫
          ℱ.map (IsOpenMap.functorMap HU HV le).op))
⊢ (obj (pullback C f) ℱ).map (homOfLE le).op ≫
      Iso.hom
        (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
          (colimitOfDiagramTerminal
            {
              lift :=
                fun
                  (s :
                    Cone (C := CostructuredArrow (Opens.map f).op (op U))
                      (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
              fac := ⋯, uniq := ⋯ :
              IsLimit (C := CostructuredArrow (Opens.map f).op (op U))
                (asEmptyCone
                  (CostructuredArrow.mk (Y :=
                    op
                      {
                        carrier :=
                          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v)))
                                  (CC := carrier) f) :
                              (a : (↑X : Type v)) → (↑Y : Type v)) ''
                            (↑U : Set (↑X : Type v)),
                        is_open' := HU : Opens (↑Y : Type v) })
                    (homOfLE ⋯).op)) }
            (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))) =
    Iso.hom
        (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op V))
          (colimitOfDiagramTerminal
            {
              lift :=
                fun
                  (s :
                    Cone (C := CostructuredArrow (Opens.map f).op (op V))
                      (empty (CostructuredArrow (Opens.map f).op (op V)))) ↦
                CostructuredArrow.homMk (homOfLE ⋯).op ⋯,
              fac := ⋯, uniq := ⋯ }
            (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ))) ≫
      ℱ.map (IsOpenMap.functorMap HU HV le).op

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

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