Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U V : Opens ↑X
HU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
HV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)
le : U ≤ V
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)).ι.app
        j ≫
      ((pullback C f).obj ℱ).map (homOfLE le).op ≫
        (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
            (colimitOfDiagramTerminal { lift := fun s ↦ CostructuredArrow.homMk (homOfLE ⋯).op ⋯, fac := ⋯, uniq := ⋯ }
              (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))).hom =
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)).ι.app
        j ≫
      (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op V))
            (colimitOfDiagramTerminal { lift := fun s ↦ CostructuredArrow.homMk (homOfLE ⋯).op ⋯, fac := ⋯, uniq := ⋯ }
              (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ))).hom ≫
        ℱ.map (IsOpenMap.functorMap HU HV le).op

Tactic:
have eq : ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ)
      ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
      (op V)).ι.app j ≫ ((pullback C f).obj ℱ).map (homOfLE le).op =
      ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ)
      ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
      (op U)).ι.app ((CostructuredArrow.map (homOfLE le).op).obj j) := by cat_disch

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U V : Opens ↑X
HU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
HV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)
le : U ≤ V
j : CostructuredArrow (Opens.map f).op (op V)
eq :
  ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)).ι.app
        j ≫
      ((pullback C f).obj ℱ).map (homOfLE le).op =
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
            (op U)).ι.app
      ((CostructuredArrow.map (homOfLE le).op).obj j)
⊢ ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)).ι.app
        j ≫
      ((pullback C f).obj ℱ).map (homOfLE le).op ≫
        (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
            (colimitOfDiagramTerminal { lift := fun s ↦ CostructuredArrow.homMk (homOfLE ⋯).op ⋯, fac := ⋯, uniq := ⋯ }
              (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))).hom =
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension ℱ) ((Opens.map f).op.leftKanExtensionUnit ℱ)).coconeAt
              (op V)).ι.app
        j ≫
      (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op V))
            (colimitOfDiagramTerminal { lift := fun s ↦ CostructuredArrow.homMk (homOfLE ⋯).op ⋯, fac := ⋯, uniq := ⋯ }
              (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ))).hom ≫
        ℱ.map (IsOpenMap.functorMap HU HV le).op