-- 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
j : CostructuredArrow (Opens.map f).op (op V)
eq :
  Eq (α := obj (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ) j ⟶ (obj (pullback C f) ℱ).obj (op U))
    (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 :
      obj (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ) j ⟶ (obj (pullback C f) ℱ).obj (op U))
    (NatTrans.app
        (Cocone.ι
          ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
            (op U)))
        (obj (CostructuredArrow.map (homOfLE le).op) j) :
      obj (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ) (obj (CostructuredArrow.map (homOfLE le).op) j) ⟶
        obj
          (obj (Functor.const (CostructuredArrow (Opens.map f).op (op U)))
            (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))))
          (obj (CostructuredArrow.map (homOfLE le).op) j))
rw :
  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 }))
    (CategoryTheory.Functor.map (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ)
        (IsTerminal.from
          {
            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 := ⋯ }
          (obj (CostructuredArrow.map (homOfLE le).op) j)) :
      obj (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ) (obj (CostructuredArrow.map (homOfLE le).op) j) ⟶
        obj (C := CostructuredArrow (Opens.map f).op (op U)) (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ)
          (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 })
            (homOfLE ⋯).op))
    (CategoryTheory.Functor.map (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ)
        (IsTerminal.from
          {
            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 := ⋯ }
          j) ≫
      ℱ.map (IsOpenMap.functorMap HU HV le).op)
⊢ 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))

Tactic:
rw [Limits.IsColimit.comp_coconePointUniqueUpToIso_hom_assoc, reassoc_of% eq,
    Limits.IsColimit.comp_coconePointUniqueUpToIso_hom,
    Limits.coconeOfDiagramTerminal_ι_app, Limits.coconeOfDiagramTerminal_ι_app]

Premises used: CategoryTheory.Limits.coconeOfDiagramTerminal_ι_app, CategoryTheory.Limits.IsColimit.comp_coconePointUniqueUpToIso_hom, CategoryTheory.Limits.IsColimit.comp_coconePointUniqueUpToIso_hom_assoc