awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
⊢ s.pt.presheaf.obj U ⟶
    obj
      (obj (C := Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
        (pushforward C (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)) :
          Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)) ⥤
            Presheaf.{v, v, u} C (Cocone.pt (F := F ⋙ forget C) (mapCocone (forget C) s)))
        (limit
            (Functor.leftOp (pushforwardDiagramToColimit F) :
              Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))) :
          Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))))
      U
have refine_1 : (j : Jᵒᵖ) →
  obj.{v', v, u', u} (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U)) j ⟶
    (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
          (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
          (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
              (obj (Functor.op (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)))) U) :
            CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)).obj
      j
Tactic: refine
    limit.lift _
        { pt := s.pt.presheaf.obj U
          π :=
            { app := fun j => ?_
              naturality := fun j j' f => ?_ } } ≫
      (limitObjIsoLimitCompEvaluation _ _).inv
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
refine_1 :
  (j : Jᵒᵖ) →
    obj.{v', v, u', u} (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U)) j ⟶
      (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
            (Functor.leftOp (pushforwardDiagramToColimit F) :
              Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
            (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
                (obj (Functor.op (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)))) U) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)).obj
        j
⊢ s.pt.presheaf.obj U ⟶
    obj
      (obj (C := Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
        (pushforward C (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)) :
          Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)) ⥤
            Presheaf.{v, v, u} C (Cocone.pt (F := F ⋙ forget C) (mapCocone (forget C) s)))
        (limit
            (Functor.leftOp (pushforwardDiagramToColimit F) :
              Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))) :
          Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))))
      U
have refine_2 : ∀ (j j' : Jᵒᵖ) (f : j ⟶ j'),
  map (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U)) f ≫
      CategoryStruct.comp (Y := (obj (pushforward C (Hom.base (s.ι.app (unop j')))) (F.obj (unop j')).presheaf).obj U)
        (NatTrans.app (Hom.c (s.ι.app (unop j'))) U) ((F.obj (unop j')).presheaf.map (eqToHom sorry)) =
    CategoryStruct.comp (Y := (obj (pushforward C (Hom.base (s.ι.app (unop j)))) (F.obj (unop j)).presheaf).obj U)
        (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom sorry)) ≫
      (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
            (Functor.leftOp (pushforwardDiagramToColimit F) :
              Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
            (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
                (obj (Functor.op (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)))) U) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)).map
        f
Tactic: refine
    limit.lift _
        { pt := s.pt.presheaf.obj U
          π :=
            { app := fun j => ?_
              naturality := fun j j' f => ?_ } } ≫
      (limitObjIsoLimitCompEvaluation _ _).inv
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
j : Jᵒᵖ
⊢ Eq (α := (Opens (↑(↑(F.obj (unop j)) : TopCat) : Type v))ᵒᵖ)
    (obj (Functor.op (Opens.map (Hom.base (s.ι.app (unop j))))) U)
    (obj (C := (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)
      (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) (unop j))))
      (obj (Functor.op (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)))) U))
have refine_1 : op (obj (Opens.map (Hom.base (s.ι.app (unop j)))) (unop U)) =
  op
    (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
      (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))
Tactic: dsimp
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
j j' : Jᵒᵖ
f : j ⟶ j'
⊢ map (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U)) f ≫
      CategoryStruct.comp (Y := (obj (pushforward C (Hom.base (s.ι.app (unop j')))) (F.obj (unop j')).presheaf).obj U)
        (NatTrans.app (Hom.c (s.ι.app (unop j'))) U) ((F.obj (unop j')).presheaf.map (eqToHom ⋯)) =
    CategoryStruct.comp (Y := (obj (pushforward C (Hom.base (s.ι.app (unop j)))) (F.obj (unop j)).presheaf).obj U)
        (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom ⋯)) ≫
      (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
            (Functor.leftOp (pushforwardDiagramToColimit F) :
              Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
            (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
                (obj (Functor.op (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)))) U) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)).map
        f
have refine_2 : Eq (α :=
  s.pt.presheaf.obj U ⟶
    (F.obj (unop j')).presheaf.obj
      (op
        (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j')))
          (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))
  (𝟙 (s.pt.presheaf.obj U) ≫
    CategoryStruct.comp (Y :=
      (F.obj (unop j')).presheaf.obj (op (obj (Opens.map (Hom.base (s.ι.app (unop j')))) (unop U))))
      (NatTrans.app (Hom.c (s.ι.app (unop j'))) U) ((F.obj (unop j')).presheaf.map (eqToHom sorry)))
  (CategoryStruct.comp (Y :=
      (F.obj (unop j)).presheaf.obj (op (obj (Opens.map (Hom.base (s.ι.app (unop j)))) (unop U))))
      (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom sorry)) ≫
    (F.map f.unop).c.app
        (op
          (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
            (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) ≫
      𝟙
          ((F.obj (unop j')).presheaf.obj
            (op
              ((Opens.map (F.map f.unop).base).obj
                (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
                  (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
                  (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))) ≫
        NatTrans.app (Iso.hom (pushforwardEq sorry (F.obj (unop j')).presheaf))
          (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
Tactic: dsimp
-- 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
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
j j' : Jᵒᵖ
f : j ⟶ j'
⊢ Eq (α :=
    s.pt.presheaf.obj U ⟶
      (F.obj (unop j')).presheaf.obj
        (op
          (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j')))
            (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))
    (𝟙 (s.pt.presheaf.obj U) ≫
      ((F.map f.unop ≫ s.ι.app (unop j)).c.app U ≫ (F.obj (unop j')).presheaf.map (eqToHom ⋯)) ≫
        (F.obj (unop j')).presheaf.map (eqToHom ⋯))
    (CategoryStruct.comp (Y :=
        (F.obj (unop j)).presheaf.obj (op (obj (Opens.map (Hom.base (s.ι.app (unop j)))) (unop U))))
        (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom ⋯)) ≫
      (F.map f.unop).c.app
          (op
            (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
              (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
              (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) ≫
        𝟙
            ((F.obj (unop j')).presheaf.obj
              (op
                ((Opens.map (F.map f.unop).base).obj
                  (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
                    (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
                    (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj (unop j')).presheaf))
            (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
have w : Eq (α := Opens (↑((F ⋙ forget C).obj (unop j)) : Type v))
  (obj
    (Opens.map
      (CategoryStruct.comp (Y := Limits.colimit (F ⋙ forget C)) (colimit.ι (F ⋙ forget C) (unop j))
        (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))))
    (unop U))
  (obj (Opens.map (NatTrans.app (Cocone.ι (mapCocone (forget C) s)) (unop j))) (unop U))
Tactic: have w :=
      Functor.congr_obj
        (congr_arg Opens.map (colimit.ι_desc ((PresheafedSpace.forget C).mapCocone s) (unop j)))
        (unop U)
-- 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
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
j j' : Jᵒᵖ
f : j ⟶ j'
w :
  Eq (α := Opens (↑((F ⋙ forget C).obj (unop j)) : Type v))
    (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
      (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))
    (obj (Opens.map (NatTrans.app (Cocone.ι (mapCocone (forget C) s)) (unop j))) (unop U))
⊢ Eq (α :=
    s.pt.presheaf.obj U ⟶
      (F.obj (unop j')).presheaf.obj
        (op
          (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j')))
            (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))
    (𝟙 (s.pt.presheaf.obj U) ≫
      ((F.map f.unop ≫ s.ι.app (unop j)).c.app U ≫ (F.obj (unop j')).presheaf.map (eqToHom ⋯)) ≫
        (F.obj (unop j')).presheaf.map (eqToHom ⋯))
    (CategoryStruct.comp (Y :=
        (F.obj (unop j)).presheaf.obj (op (obj (Opens.map (Hom.base (s.ι.app (unop j)))) (unop U))))
        (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom ⋯)) ≫
      (F.map f.unop).c.app
          (op
            (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
              (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
              (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) ≫
        𝟙
            ((F.obj (unop j')).presheaf.obj
              (op
                ((Opens.map (F.map f.unop).base).obj
                  (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
                    (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
                    (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj (unop j')).presheaf))
            (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
have w₁ : Eq (α := (Opens (↑((F ⋙ forget C).obj (unop j)) : Type v))ᵒᵖ)
  (op
    (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
      (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
  (op (obj (Opens.map (NatTrans.app (Cocone.ι (mapCocone (forget C) s)) (unop j))) (unop U)))
Tactic: replace w := congr_arg op w
Cleared hypotheses: w
-- 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
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
U : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
j j' : Jᵒᵖ
f : j ⟶ j'
w :
  Eq (α := (Opens (↑((F ⋙ forget C).obj (unop j)) : Type v))ᵒᵖ)
    (op
      (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
        (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
    (op (obj (Opens.map (NatTrans.app (Cocone.ι (mapCocone (forget C) s)) (unop j))) (unop U)))
⊢ Eq (α :=
    s.pt.presheaf.obj U ⟶
      (F.obj (unop j')).presheaf.obj
        (op
          (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j')))
            (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))
    (𝟙 (s.pt.presheaf.obj U) ≫
      ((F.map f.unop ≫ s.ι.app (unop j)).c.app U ≫ (F.obj (unop j')).presheaf.map (eqToHom ⋯)) ≫
        (F.obj (unop j')).presheaf.map (eqToHom ⋯))
    (CategoryStruct.comp (Y :=
        (F.obj (unop j)).presheaf.obj (op (obj (Opens.map (Hom.base (s.ι.app (unop j)))) (unop U))))
        (NatTrans.app (Hom.c (s.ι.app (unop j))) U) ((F.obj (unop j)).presheaf.map (eqToHom ⋯)) ≫
      (F.map f.unop).c.app
          (op
            (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
              (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
              (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) ≫
        𝟙
            ((F.obj (unop j')).presheaf.obj
              (op
                ((Opens.map (F.map f.unop).base).obj
                  (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))
                    (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
                    (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))))) ≫
          NatTrans.app (Iso.hom (pushforwardEq ⋯ (F.obj (unop j')).presheaf))
            (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))))
have w' : (F.map f.unop).c.app
    (op
      (obj (C := Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v)) (Opens.map (colimit.ι (F ⋙ forget C) (unop j)))
        (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) =
  (F.obj (unop j)).presheaf.map (eqToHom w) ≫
    (F.map f.unop).c.app (op (obj (Opens.map (NatTrans.app (Cocone.ι (mapCocone (forget C) s)) (unop j))) (unop U))) ≫
      (obj (pushforward C (F.map f.unop).base) (F.obj (unop j')).presheaf).map (eqToHom sorry)
Tactic: have w' := NatTrans.congr (F.map f.unop).c w
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
j : J
⊢ (colimitCocone F).ι.app j ≫ desc F s = s.ι.app j
have w : ∀ (U : (↑(↑(F.obj j) : TopCat) : Type v)),
  Eq (α := (↑(↑s.pt : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            ((colimitCocone F).ι.app j ≫ desc F s).base :
          (a : (↑(↑(F.obj j) : TopCat) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        U :
      (↑(↑s.pt : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (s.ι.app j).base :
          (a : (↑(↑(F.obj j) : TopCat) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        U :
      (↑(↑s.pt : TopCat) : Type v))
Tactic: ext U
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
j : J
w :
  ∀ (U : (↑(↑(F.obj j) : TopCat) : Type v)),
    Eq (α := (↑(↑s.pt : TopCat) : Type v))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              ((colimitCocone F).ι.app j ≫ desc F s).base :
            (a : (↑(↑(F.obj j) : TopCat) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
          U :
        (↑(↑s.pt : TopCat) : Type v))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (s.ι.app j).base :
            (a : (↑(↑(F.obj j) : TopCat) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
          U :
        (↑(↑s.pt : TopCat) : Type v))
⊢ (colimitCocone F).ι.app j ≫ desc F s = s.ι.app j
have h : ∀ (U : Opens (↑(↑s.pt : TopCat) : Type v)),
  Eq (α := s.pt.presheaf.obj (op U) ⟶ ((Opens.map (s.ι.app j).base).op ⋙ (F.obj j).presheaf).obj (op U))
    (NatTrans.app (((colimitCocone F).ι.app j ≫ desc F s).c ≫ whiskerRight (eqToHom sorry) (F.obj j).presheaf) (op U))
    ((s.ι.app j).c.app (op U))
Tactic: ext U
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
j : J
U : Opens (↑(↑s.pt : TopCat) : Type v)
⊢ Eq (α := s.pt.presheaf.obj (op U) ⟶ ((Opens.map (s.ι.app j).base).op ⋙ (F.obj j).presheaf).obj (op U))
    (NatTrans.app (((colimitCocone F).ι.app j ≫ desc F s).c ≫ whiskerRight (eqToHom ⋯) (F.obj j).presheaf) (op U))
    ((s.ι.app j).c.app (op U))
have h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: simp only [op_obj, desc, descCApp, Presheaf.comp_app, comp_c_app, colimitCocone_ι_app_c, 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
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
⊢ m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
have apply : ∀ (j : J),
  colimit.ι (F ⋙ forget C) j ≫ m.base =
    CategoryStruct.comp (Y := Limits.colimit (F ⋙ forget C)) (colimit.ι (F ⋙ forget C) j)
      (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))
Tactic: apply colimit.hom_ext fun j => ?_
-- 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)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
j : J
⊢ colimit.ι (F ⋙ forget C) j ≫ m.base =
    CategoryStruct.comp (Y := Limits.colimit (F ⋙ forget C)) (colimit.ι (F ⋙ forget C) j)
      (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))
have w₁ : ∀ (x : (↑((F ⋙ forget C).obj j) : Type v)),
  Eq (α := (↑(↑s.pt : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (colimit.ι (F ⋙ forget C) j ≫ m.base) :
          (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        x :
      (↑(↑s.pt : TopCat) : Type v))
    (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑s.pt : TopCat))
              (CategoryStruct.comp (Y := Limits.colimit (F ⋙ forget C)) (colimit.ι (F ⋙ forget C) j)
                  (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)) :
                (F ⋙ forget C).obj j ⟶ (↑s.pt : TopCat)) :
            C((↑((F ⋙ forget C).obj j) : Type v), (↑(↑s.pt : TopCat) : Type v))) :
          (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        x :
      (↑(↑s.pt : TopCat) : Type v))
Tactic: ext