l3lab/ntp-mathlib huggingface dataset:
/-- Auxiliary definition for `AlgebraicGeometry.PresheafedSpace.instHasColimits`.
-/
def colimitCoconeIsColimit (F : J ⥤ PresheafedSpace.{_, _, v} C) :
    IsColimit (colimitCocone F) where
  desc s := desc F s
  fac s := desc_fac F s
  uniq s m w := by
    -- We need to use the identity on the continuous maps twice, so we prepare that first:
    have t :
      m.base =
        colimit.desc (F ⋙ PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) := by
      dsimp
      ext j
      rw [colimit.ι_desc, mapCocone_ι_app, ← w j]
      simp
    ext : 1
    · exact t
    · refine NatTrans.ext _ _ (funext fun U => limit_obj_ext fun j => ?_)
      dsimp only [colimitCocone_pt, colimit_carrier, leftOp_obj, pushforwardDiagramToColimit_obj,
        comp_obj, forget_obj, unop_op, op_obj, desc, colimit_presheaf, descCApp, mapCocone_pt,
        pushforwardObj_obj, const_obj_obj, id_eq, evaluation_obj_obj, Eq.ndrec, eq_mpr_eq_cast]
      rw [NatTrans.comp_app, whiskerRight_app]
      simp only [pushforwardObj_obj, op_obj, comp_obj, eqToHom_app, eqToHom_map, assoc,
        limitObjIsoLimitCompEvaluation_inv_π_app, limit.lift_π]
      rw [PresheafedSpace.congr_app (w (unop j)).symm U]
      dsimp
      

Proof state:
case h
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
t : m.base = colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)
U : (Opens ↑↑s.pt)ᵒᵖ
j : Jᵒᵖ
⊢ m.c.app U ≫
      eqToHom
          (_ :
            (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =
              (limit (pushforwardDiagramToColimit F).leftOp).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop))) ≫
        (limit.π (pushforwardDiagramToColimit F).leftOp j).app
          (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)) =
    ((m.c.app U ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map m.base).obj U.unop))) ≫
        (F.obj j.unop).presheaf.map
          (eqToHom
            (_ :
              op ((Opens.map ((colimitCocone F).ι.app j.unop ≫ m).base).obj U.unop) =
                op ((Opens.map (s.ι.app j.unop).base).obj U.unop)))) ≫
      eqToHom
        (_ :
          (F.obj j.unop).presheaf.obj (op ((Opens.map (s.ι.app j.unop).base).obj U.unop)) =
            (F.obj j.unop).presheaf.obj
              (op
                ((Opens.map (colimit.ι (F ⋙ forget C) j.unop)).obj
                  ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop))))

Next tactic:
have w := congr_arg op (Functor.congr_obj (congr_arg Opens.map t) (unop U))
/-- Auxiliary definition for `AlgebraicGeometry.PresheafedSpace.instHasColimits`.
-/
def colimitCoconeIsColimit (F : J ⥤ PresheafedSpace.{_, _, v} C) :
    IsColimit (colimitCocone F) where
  desc s := desc F s
  fac s := desc_fac F s
  uniq s m w := by
    -- We need to use the identity on the continuous maps twice, so we prepare that first:
    have t :
      m.base =
        colimit.desc (F ⋙ PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) := by
      dsimp
      ext j
      rw [colimit.ι_desc, mapCocone_ι_app, ← w j]
      simp
    ext : 1
    · exact t
    · refine NatTrans.ext _ _ (funext fun U => limit_obj_ext fun j => ?_)
      dsimp only [colimitCocone_pt, colimit_carrier, leftOp_obj, pushforwardDiagramToColimit_obj,
        comp_obj, forget_obj, unop_op, op_obj, desc, colimit_presheaf, descCApp, mapCocone_pt,
        pushforwardObj_obj, const_obj_obj, id_eq, evaluation_obj_obj, Eq.ndrec, eq_mpr_eq_cast]
      rw [NatTrans.comp_app, whiskerRight_app]
      simp only [pushforwardObj_obj, op_obj, comp_obj, eqToHom_app, eqToHom_map, assoc,
        limitObjIsoLimitCompEvaluation_inv_π_app, limit.lift_π]
      rw [PresheafedSpace.congr_app (w (unop j)).symm U]
      dsimp
      have w := congr_arg op (Functor.congr_obj (congr_arg Opens.map t) (unop U))
      

Proof state:
case h
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w✝ : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
t : m.base = colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)
U : (Opens ↑↑s.pt)ᵒᵖ
j : Jᵒᵖ
w :
  op ((Opens.map m.base).obj U.unop) =
    op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)
⊢ m.c.app U ≫
      eqToHom
          (_ :
            (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =
              (limit (pushforwardDiagramToColimit F).leftOp).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop))) ≫
        (limit.π (pushforwardDiagramToColimit F).leftOp j).app
          (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)) =
    ((m.c.app U ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map m.base).obj U.unop))) ≫
        (F.obj j.unop).presheaf.map
          (eqToHom
            (_ :
              op ((Opens.map ((colimitCocone F).ι.app j.unop ≫ m).base).obj U.unop) =
                op ((Opens.map (s.ι.app j.unop).base).obj U.unop)))) ≫
      eqToHom
        (_ :
          (F.obj j.unop).presheaf.obj (op ((Opens.map (s.ι.app j.unop).base).obj U.unop)) =
            (F.obj j.unop).presheaf.obj
              (op
                ((Opens.map (colimit.ι (F ⋙ forget C) j.unop)).obj
                  ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop))))

Next tactic:
rw [NatTrans.congr (limit.π (pushforwardDiagramToColimit F).leftOp j) w]
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
  ⟨fun {F} => preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F) <| by
    

Proof state:
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
⊢ IsColimit ((forget C).mapCocone (colimitCocone F))

Next tactic:
apply IsColimit.ofIsoColimit (colimit.isColimit _)
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
  ⟨fun {F} => preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F) <| by
    apply IsColimit.ofIsoColimit (colimit.isColimit _)
    

Proof state:
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
⊢ colimit.cocone (F ⋙ forget C) ≅ (forget C).mapCocone (colimitCocone F)

Next tactic:
fapply Cocones.ext
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
  ⟨fun {F} => preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F) <| by
    apply IsColimit.ofIsoColimit (colimit.isColimit _)
    fapply Cocones.ext
    · 

Proof state:
case φ
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
⊢ (colimit.cocone (F ⋙ forget C)).pt ≅ ((forget C).mapCocone (colimitCocone F)).pt

Next tactic:
rfl
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
  ⟨fun {F} => preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F) <| by
    apply IsColimit.ofIsoColimit (colimit.isColimit _)
    fapply Cocones.ext
    · rfl
    · 

Proof state:
case w
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
⊢ autoParam
    (∀ (j : J),
      (colimit.cocone (F ⋙ forget C)).ι.app j ≫ (Iso.refl (colimit.cocone (F ⋙ forget C)).pt).hom =
        ((forget C).mapCocone (colimitCocone F)).ι.app j)
    _auto✝

Next tactic:
intro j
instance : PreservesColimitsOfShape J (PresheafedSpace.forget.{u, v, v} C) :=
  ⟨fun {F} => preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F) <| by
    apply IsColimit.ofIsoColimit (colimit.isColimit _)
    fapply Cocones.ext
    · rfl
    · intro j
      

Proof state:
case w
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
j : J
⊢ (colimit.cocone (F ⋙ forget C)).ι.app j ≫ (Iso.refl (colimit.cocone (F ⋙ forget C)).pt).hom =
    ((forget C).mapCocone (colimitCocone F)).ι.app j

Next tactic:
simp
/-- The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces.
-/
instance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C) where
  preservesColimitsOfShape {J 𝒥} :=
    { preservesColimit := fun {F} =>
        preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)
          (by 

Proof state:
J✝ : Type u'
inst✝⁵ : Category.{v', u'} J✝
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J✝ TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape J✝ᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape J✝ᵒᵖ C
inst✝ : HasLimits C
J : Type v
𝒥 : Category.{v, v} J
F : J ⥤ PresheafedSpace C
⊢ IsColimit ((forget C).mapCocone (colimitCocone F))

Next tactic:
apply IsColimit.ofIsoColimit (colimit.isColimit _)
/-- The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces.
-/
instance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C) where
  preservesColimitsOfShape {J 𝒥} :=
    { preservesColimit := fun {F} =>
        preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)
          (by apply IsColimit.ofIsoColimit (colimit.isColimit _)
              

Proof state:
J✝ : Type u'
inst✝⁵ : Category.{v', u'} J✝
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J✝ TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape J✝ᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape J✝ᵒᵖ C
inst✝ : HasLimits C
J : Type v
𝒥 : Category.{v, v} J
F : J ⥤ PresheafedSpace C
⊢ colimit.cocone (F ⋙ forget C) ≅ (forget C).mapCocone (colimitCocone F)

Next tactic:
fapply Cocones.ext
/-- The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces.
-/
instance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C) where
  preservesColimitsOfShape {J 𝒥} :=
    { preservesColimit := fun {F} =>
        preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)
          (by apply IsColimit.ofIsoColimit (colimit.isColimit _)
              fapply Cocones.ext
              · 

Proof state:
case φ
J✝ : Type u'
inst✝⁵ : Category.{v', u'} J✝
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J✝ TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape J✝ᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape J✝ᵒᵖ C
inst✝ : HasLimits C
J : Type v
𝒥 : Category.{v, v} J
F : J ⥤ PresheafedSpace C
⊢ (colimit.cocone (F ⋙ forget C)).pt ≅ ((forget C).mapCocone (colimitCocone F)).pt

Next tactic:
rfl
/-- The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces.
-/
instance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C) where
  preservesColimitsOfShape {J 𝒥} :=
    { preservesColimit := fun {F} =>
        preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)
          (by apply IsColimit.ofIsoColimit (colimit.isColimit _)
              fapply Cocones.ext
              · rfl
              · 

Proof state:
case w
J✝ : Type u'
inst✝⁵ : Category.{v', u'} J✝
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J✝ TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape J✝ᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape J✝ᵒᵖ C
inst✝ : HasLimits C
J : Type v
𝒥 : Category.{v, v} J
F : J ⥤ PresheafedSpace C
⊢ autoParam
    (∀ (j : J),
      (colimit.cocone (F ⋙ forget C)).ι.app j ≫ (Iso.refl (colimit.cocone (F ⋙ forget C)).pt).hom =
        ((forget C).mapCocone (colimitCocone F)).ι.app j)
    _auto✝

Next tactic:
intro j
/-- The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces.
-/
instance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C) where
  preservesColimitsOfShape {J 𝒥} :=
    { preservesColimit := fun {F} =>
        preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)
          (by apply IsColimit.ofIsoColimit (colimit.isColimit _)
              fapply Cocones.ext
              · rfl
              · intro j
                

Proof state:
case w
J✝ : Type u'
inst✝⁵ : Category.{v', u'} J✝
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J✝ TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape J✝ᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape J✝ᵒᵖ C
inst✝ : HasLimits C
J : Type v
𝒥 : Category.{v, v} J
F : J ⥤ PresheafedSpace C
j : J
⊢ (colimit.cocone (F ⋙ forget C)).ι.app j ≫ (Iso.refl (colimit.cocone (F ⋙ forget C)).pt).hom =
    ((forget C).mapCocone (colimitCocone F)).ι.app j

Next tactic:
simp
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  

Proof state:
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
⊢ (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U)

Next tactic:
refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  

Proof state:
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
⊢ ((colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base _*
          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt.presheaf).obj
      (op U) ≅
    limit (componentwiseDiagram F U)

Next tactic:
refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  

Proof state:
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
⊢ (pushforwardDiagramToColimit F).leftOp ⋙
      (evaluation (Opens ↑↑{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)ᵒᵖ C).obj
        ((Opens.map
                (colimit.isoColimitCocone
                        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj
          (op U)) ≅
    componentwiseDiagram F U

Next tactic:
fapply NatIso.ofComponents
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  fapply NatIso.ofComponents
  · 

Proof state:
case app
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
⊢ (X : Jᵒᵖ) →
    ((pushforwardDiagramToColimit F).leftOp ⋙
            (evaluation (Opens ↑↑{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)ᵒᵖ
                  C).obj
              ((Opens.map
                      (colimit.isoColimitCocone
                              { cocone := colimitCocone F,
                                isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj
                (op U))).obj
        X ≅
      (componentwiseDiagram F U).obj X

Next tactic:
intro X
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  fapply NatIso.ofComponents
  · intro X
    

Proof state:
case app
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
X : Jᵒᵖ
⊢ ((pushforwardDiagramToColimit F).leftOp ⋙
          (evaluation (Opens ↑↑{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)ᵒᵖ C).obj
            ((Opens.map
                    (colimit.isoColimitCocone
                            { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj
              (op U))).obj
      X ≅
    (componentwiseDiagram F U).obj X

Next tactic:
refine' (F.obj (unop X)).presheaf.mapIso (eqToIso _)
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  fapply NatIso.ofComponents
  · intro X
    refine' (F.obj (unop X)).presheaf.mapIso (eqToIso _)
    

Proof state:
case app
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
X : Jᵒᵖ
⊢ (Opens.map (colimit.ι (F ⋙ forget C) X.unop)).op.obj
      ((Opens.map
              (colimit.isoColimitCocone
                      { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj
        (op U)) =
    op ((Opens.map (colimit.ι F X.unop).base).obj U)

Next tactic:
simp only [Functor.op_obj, unop_op, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
      Set.preimage_preimage]
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  fapply NatIso.ofComponents
  · intro X
    refine' (F.obj (unop X)).presheaf.mapIso (eqToIso _)
    simp only [Functor.op_obj, unop_op, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
      Set.preimage_preimage]
    

Proof state:
case app
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
X : Jᵒᵖ
⊢ (fun x =>
        (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base
          ((colimit.ι (F ⋙ forget C) X.unop) x)) ⁻¹'
      ↑U =
    ⇑(colimit.ι F X.unop).base ⁻¹' ↑U

Next tactic:
refine congr_arg (Set.preimage . U.1) (funext fun x => ?_)
/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits.
-/
def colimitPresheafObjIsoComponentwiseLimit (F : J ⥤ PresheafedSpace.{_, _, v} C) [HasColimit F]
    (U : Opens (Limits.colimit F).carrier) :
    (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U) := by
  refine'
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      _
  refine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)
  fapply NatIso.ofComponents
  · intro X
    refine' (F.obj (unop X)).presheaf.mapIso (eqToIso _)
    simp only [Functor.op_obj, unop_op, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
      Set.preimage_preimage]
    refine congr_arg (Set.preimage . U.1) (funext fun x => ?_)
    

Proof state:
case app
J : Type u'
inst✝⁵ : Category.{v', u'} J
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasColimitsOfShape J TopCat
inst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝¹ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
inst✝ : HasColimit F
U : Opens ↑↑(Limits.colimit F)
X : Jᵒᵖ
x : (CategoryTheory.forget TopCat).obj ((F ⋙ forget C).obj X.unop)
⊢ (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base
      ((colimit.ι (F ⋙ forget C) X.unop) x) =
    (colimit.ι F X.unop).base x

Next tactic:
erw [← comp_app]