-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
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
⊢ (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
have 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 sorry).op sorry,
                fac := sorry, uniq := sorry :
                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 sorry).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 sorry).op sorry,
                fac := sorry, uniq := sorry }
              (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 ↦ ?_)