awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
s : PullbackCone (Z := D.glued) (D.ι i) (D.ι j)
⊢ Eq (α := s.pt ⟶ D.glued) (lift (D.f i j) s.fst ⋯ ≫ PullbackCone.fst (D.vPullbackCone i j) ≫ D.ι i) (s.fst ≫ D.ι i)
have refine_3 : Eq (α := s.pt ⟶ D.glued) (s.fst ≫ D.ι i) (s.fst ≫ D.ι i)
Tactic: erw [IsOpenImmersion.lift_fac_assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
s : PullbackCone (Z := D.glued) (D.ι i) (D.ι j)
m : s.pt ⟶ Cone.pt (F := cospan (Z := D.glued) (D.ι i) (D.ι j)) (D.vPullbackCone i j)
e₁ : m ≫ PullbackCone.fst (D.vPullbackCone i j) = s.fst
h : m ≫ PullbackCone.snd (D.vPullbackCone i j) = s.snd
⊢ m = lift (D.f i j) s.fst ⋯
have refine_4 : m ≫ D.f i j = lift (D.f i j) s.fst sorry ≫ D.f i j
Tactic: rw [← cancel_mono (D.f i j)]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
s : PullbackCone (Z := D.glued) (D.ι i) (D.ι j)
m : s.pt ⟶ Cone.pt (F := cospan (Z := D.glued) (D.ι i) (D.ι j)) (D.vPullbackCone i j)
e₁ : m ≫ PullbackCone.fst (D.vPullbackCone i j) = s.fst
h : m ≫ PullbackCone.snd (D.vPullbackCone i j) = s.snd
⊢ m ≫ D.f i j = lift (D.f i j) s.fst ⋯ ≫ D.f i j
have refine_4 : m ≫ D.f i j = s.fst
Tactic: simp only [lift_fac]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
⊢ PresheafedSpace.IsOpenImmersion (Y := D.glued.toPresheafedSpace) (InducedCategory.Hom.hom (D.ι i))
have rw : PresheafedSpace.IsOpenImmersion (Y := D.glued.toPresheafedSpace)
  (CategoryStruct.comp (Y := D.toPresheafedSpaceGlueData.glued)
      (D.toPresheafedSpaceGlueData.ι i : D.toPresheafedSpaceGlueData.U i ⟶ D.toPresheafedSpaceGlueData.glued)
      (Iso.inv D.isoPresheafedSpace : D.toPresheafedSpaceGlueData.glued ⟶ D.glued.toPresheafedSpace) :
    D.toPresheafedSpaceGlueData.U i ⟶ D.glued.toPresheafedSpace)
Tactic: rw [← D.ι_isoPresheafedSpace_inv]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
⊢ PresheafedSpace.IsOpenImmersion (Y := D.glued.toPresheafedSpace)
    (CategoryStruct.comp (Y := D.toPresheafedSpaceGlueData.glued)
        (D.toPresheafedSpaceGlueData.ι i : D.toPresheafedSpaceGlueData.U i ⟶ D.toPresheafedSpaceGlueData.glued)
        (Iso.inv D.isoPresheafedSpace : D.toPresheafedSpaceGlueData.glued ⟶ D.glued.toPresheafedSpace) :
      D.toPresheafedSpaceGlueData.U i ⟶ D.glued.toPresheafedSpace)
have this : PresheafedSpace.IsOpenImmersion (Y := D.toPresheafedSpaceGlueData.glued) (D.toPresheafedSpaceGlueData.ι i)
Tactic: have := D.toPresheafedSpaceGlueData.ιIsOpenImmersion i
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData
i : D.J
⊢ SheafedSpace.IsOpenImmersion (Y := D.glued.toSheafedSpace) (Hom.toShHom (D.ι i))
have rw : SheafedSpace.IsOpenImmersion (D.toSheafedSpaceGlueData.ι i ≫ D.isoSheafedSpace.inv)
Tactic: rw [← D.ι_isoSheafedSpace_inv]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData
i : D.J
⊢ SheafedSpace.IsOpenImmersion (D.toSheafedSpaceGlueData.ι i ≫ D.isoSheafedSpace.inv)
have H : PresheafedSpace.IsOpenImmersion (Y := D.toSheafedSpaceGlueData.glued.toPresheafedSpace)
  (InducedCategory.Hom.hom (D.toSheafedSpaceGlueData.ι i))
Tactic: apply +allowSynthFailures PresheafedSpace.IsOpenImmersion.comp
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_2 : HasColimit F
U : Opens (↑(↑(colimit F) : TopCat) : Type v)
j k : Jᵒᵖ
f : j ⟶ k
⊢ (Opens.map (F.map f.unop).base).op.obj (op (obj (Opens.map (Hom.base (colimit.ι F (unop j)))) U)) =
    op (obj (Opens.map (Hom.base (colimit.ι F (unop k)))) U)
have rw : (Opens.map (F.map f.unop).base).op.obj (op (obj (Opens.map (Hom.base (colimit.ι F (unop j)))) U)) =
  op (obj (Opens.map ((F.map f.unop).base ≫ Hom.base (colimit.ι F (unop j)))) U)
Tactic: rw [← colimit.w F f.unop, comp_base]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_2 : HasColimit F
U : Opens (↑(↑(colimit F) : TopCat) : Type v)
i j k : Jᵒᵖ
f : i ⟶ j
g : j ⟶ k
⊢ (F.map (f ≫ g).unop).c.app (op (obj (Opens.map (Hom.base (colimit.ι F (unop i)))) U)) ≫
      (F.obj (unop k)).presheaf.map (eqToHom ⋯) =
    ((F.map f.unop).c.app (op (obj (Opens.map (Hom.base (colimit.ι F (unop i)))) U)) ≫
        (F.obj (unop j)).presheaf.map (eqToHom ⋯)) ≫
      (F.map g.unop).c.app (op (obj (Opens.map (Hom.base (colimit.ι F (unop j)))) U)) ≫
        (F.obj (unop k)).presheaf.map (eqToHom ⋯)
have simp : (F.map (f ≫ g).unop).c.app (op (obj (Opens.map (Hom.base (colimit.ι F (unop i)))) U)) ≫
    (F.obj (unop k)).presheaf.map (eqToHom sorry) =
  (F.map f.unop).c.app (op (obj (Opens.map (Hom.base (colimit.ι F (unop i)))) U)) ≫
    (F.map g.unop).c.app
        ((Opens.map (F.map f.unop).base).op.obj (op (obj (Opens.map (Hom.base (colimit.ι F (unop i)))) U))) ≫
      (obj (pushforward C (F.map g.unop).base) (F.obj (unop k)).presheaf).map (eqToHom sorry) ≫
        (F.obj (unop k)).presheaf.map (eqToHom sorry)
Tactic: simp only [assoc, CategoryTheory.NatTrans.naturality_assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j : J
⊢ Eq (α :=
    Quiver.Hom (V := (Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))ᵒᵖ)
      (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
      (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)))
    (Quiver.Hom.op
      (map (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.map (𝟙 j)).c ≫
        Iso.inv (Pushforward.comp ((F ⋙ forget C).map (𝟙 j)) (colimit.ι (F ⋙ forget C) j) (F.obj j).presheaf) ≫
          Iso.hom (pushforwardEq ⋯ (F.obj j).presheaf)))
    (𝟙 (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)))
have a : Eq (α :=
  Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
    (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)))
    (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
  ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
          (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
        (a :
            op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
          unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
      (Quiver.Hom.op
        (map (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.map (𝟙 j)).c ≫
          Iso.inv (Pushforward.comp ((F ⋙ forget C).map (𝟙 j)) (colimit.ι (F ⋙ forget C) j) (F.obj j).presheaf) ≫
            Iso.hom (pushforwardEq sorry (F.obj j).presheaf))) :
    Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
  ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
          (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
        (a :
            op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
          unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
      (𝟙 (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) :
    Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
Tactic: apply (opEquiv _ _).injective
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j : J
U : Opens (↑(colimit (F ⋙ forget C)) : Type v)
⊢ Eq (α :=
    obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) (op U) ⟶
      obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) (op U))
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
        (Quiver.Hom.op
          (map (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.map (𝟙 j)).c ≫
            Iso.inv (Pushforward.comp ((F ⋙ forget C).map (𝟙 j)) (colimit.ι (F ⋙ forget C) j) (F.obj j).presheaf) ≫
              Iso.hom (pushforwardEq ⋯ (F.obj j).presheaf))))
      (op U))
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
        (𝟙 (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
      (op U))
have a : Eq.{v + 1} (α :=
  (F.obj j).presheaf.obj (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j)) U)) ⟶
    (F.obj j).presheaf.obj (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j)) U)))
  (𝟙 ((F.obj j).presheaf.obj (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j)) U))))
  (NatTrans.app (𝟙 (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) (op U))
Tactic: simp [opEquiv]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
⊢ op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) =
    op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))
have a : unop (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U)))) =
  unop (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U)))
Tactic: apply unop_injective
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
⊢ unop (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U)))) =
    unop (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U)))
have a : unop (op (obj (Opens.map ((F.map g).base ≫ colimit.ι (F ⋙ forget C) j₃)) (unop U))) =
  unop (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U)))
Tactic: rw [← Opens.map_comp_obj]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
⊢ unop (op (obj (Opens.map ((F.map g).base ≫ colimit.ι (F ⋙ forget C) j₃)) (unop U))) =
    unop (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U)))
have a : Eq (α := (↑(F.obj j₂) : TopCat) ⟶ colimit (F ⋙ forget C)) ((F.map g).base ≫ colimit.ι (F ⋙ forget C) j₃)
  (colimit.ι (F ⋙ forget C) j₂)
Tactic: congr
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (hypothesis)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
⊢ (F.map (f ≫ g)).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
      𝟙
          ((F.obj j₁).presheaf.obj
            (op ((Opens.map (F.map (f ≫ g)).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))))) ≫
        NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf)) U =
    ((F.map g).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
        𝟙
            ((F.obj j₂).presheaf.obj
              (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₂).presheaf)) U) ≫
      (F.map f).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))) ≫
        𝟙
            ((F.obj j₁).presheaf.obj
              (op ((Opens.map (F.map f).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf)) U
have this : op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) =
  op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))
Tactic: have :
      op ((Opens.map (F.map g).base).obj
          ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj U.unop)) =
        op ((Opens.map (colimit.ι (F ⋙ PresheafedSpace.forget C) j₂)).obj (unop U)) := by
      apply unop_injective
      rw [← Opens.map_comp_obj]
      congr
      exact colimit.w (F ⋙ PresheafedSpace.forget C) g
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
this :
  op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) =
    op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))
⊢ (F.map (f ≫ g)).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
      𝟙
          ((F.obj j₁).presheaf.obj
            (op ((Opens.map (F.map (f ≫ g)).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))))) ≫
        NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf)) U =
    ((F.map g).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
        𝟙
            ((F.obj j₂).presheaf.obj
              (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₂).presheaf)) U) ≫
      (F.map f).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))) ≫
        𝟙
            ((F.obj j₁).presheaf.obj
              (op ((Opens.map (F.map f).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₂)) (unop U))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf)) U
have a : Eq (α :=
  (F.obj j₃).presheaf.obj (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ⟶
    (F.obj j₁).presheaf.obj (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₁)) (unop U))))
  ((F.map g).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
    (F.map f).c.app (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U)))) ≫
      (F.obj j₁).presheaf.map (eqToHom sorry))
  ((F.map g).c.app (op (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U))) ≫
    (F.map f).c.app (op ((Opens.map (F.map g).base).obj (obj (Opens.map (colimit.ι (F ⋙ forget C) j₃)) (unop U)))) ≫
      (F.obj j₁).presheaf.map (Quiver.Hom.op ((Opens.map (F.map f).base).map (eqToHom sorry))) ≫
        (F.obj j₁).presheaf.map (eqToHom sorry))
Tactic: simp only [map_comp_c_app, pushforward_obj_obj, pushforward_map_app, comp_base,
      pushforwardEq_hom_app, op_obj, Opens.map_comp_obj, id_comp, assoc, eqToHom_map_comp,
      NatTrans.naturality_assoc, pushforward_obj_map, eqToHom_unop]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
F : J ⥤ PresheafedSpace.{v, u, v} C
j j' : J
f : j ⟶ j'
⊢ Eq (α := F.obj j ⟶ obj (obj (Functor.const J) (colimit F)) j')
    (F.map f ≫
      { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') })
    ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
      map (obj (Functor.const J) (colimit F)) f)
have w : Eq (α := (↑(F.obj j) : TopCat) ⟶ (↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat))
  (Hom.base
    (F.map f ≫
      { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') }))
  (Hom.base
    ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
      map (obj (Functor.const J) (colimit F)) f))
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
F : J ⥤ PresheafedSpace.{v, u, v} C
j j' : J
f : j ⟶ j'
w :
  Eq (α := (↑(F.obj j) : TopCat) ⟶ (↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat))
    (Hom.base
      (F.map f ≫
        { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') }))
    (Hom.base
      ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
        map (obj (Functor.const J) (colimit F)) f))
⊢ Eq (α := F.obj j ⟶ obj (obj (Functor.const J) (colimit F)) j')
    (F.map f ≫
      { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') })
    ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
      map (obj (Functor.const J) (colimit F)) f)
have h : Eq (α :=
  (obj (obj (Functor.const J) (colimit F)) j').presheaf ⟶
    Functor.op
        (Opens.map
          (Hom.base
            ({ base := colimit.ι (F ⋙ forget C) j,
                c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
              map (obj (Functor.const J) (colimit F)) f))) ⋙
      (F.obj j).presheaf)
  (CategoryStruct.comp (Y :=
    obj
      (pushforward C
        (Hom.base
          (F.map f ≫
            { base := colimit.ι (F ⋙ forget C) j',
              c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') :
              (F.obj j').Hom (obj (obj (Functor.const J) (colimit F)) j') })))
      (F.obj j).presheaf)
    (Hom.c
      (F.map f ≫
        { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') }))
    (whiskerRight (eqToHom sorry) (F.obj j).presheaf))
  (Hom.c
    ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
      map (obj (Functor.const J) (colimit F)) f))
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
F : J ⥤ PresheafedSpace.{v, u, v} C
j j' : J
f : j ⟶ j'
⊢ Eq (α := (↑(F.obj j) : TopCat) ⟶ (↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat))
    (Hom.base
      (F.map f ≫
        { base := colimit.ι (F ⋙ forget C) j', c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') }))
    (Hom.base
      ({ base := colimit.ι (F ⋙ forget C) j, c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
        map (obj (Functor.const J) (colimit F)) f))
have w : ∀ (x : (↑(↑(F.obj j) : TopCat) : Type v)),
  Eq (α := (↑(↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (Y :=
            (↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat))
            (Hom.base
              (F.map f ≫
                { base := colimit.ι (F ⋙ forget C) j',
                  c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j') })) :
          (a : (↑(↑(F.obj j) : TopCat) : Type v)) →
            (↑(↑(((Functor.const J).obj (colimit F)).obj j') : TopCat) : Type v))
        x :
      (↑(↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(F.obj j) : TopCat)) (Y := (↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat))
            (Hom.base
              ({ base := colimit.ι (F ⋙ forget C) j,
                  c := limit.π (Functor.leftOp (pushforwardDiagramToColimit F)) (op j) } ≫
                map (obj (Functor.const J) (colimit F)) f)) :
          (a : (↑(↑(F.obj j) : TopCat) : Type v)) →
            (↑(↑(((Functor.const J).obj (colimit F)).obj j') : TopCat) : Type v))
        x :
      (↑(↑(obj (obj (Functor.const J) (colimit F)) j') : TopCat) : Type v))
Tactic: ext x