awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
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))
⊢ 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:
ext
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
x : (↑((F ⋙ forget C).obj j) : Type v)
w₁ :
  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)
              (X := (F ⋙ forget C).obj j) (Y := (↑s.pt : TopCat))
              (map (forget C) ((colimitCocone F).ι.app j ≫ m) : obj (forget C) (F.obj j) ⟶ obj (forget C) s.pt) :
            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))
⊢ 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:
rw [colimit.ι_desc, mapCocone_ι_app, ← w j]

Premises used: CategoryTheory.Functor.mapCocone_ι_app, CategoryTheory.Limits.colimit.ι_desc
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
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)
              (X := (F ⋙ forget C).obj j) (Y := (↑s.pt : TopCat))
              (map (forget C) ((colimitCocone F).ι.app j ≫ m) : obj (forget C) (F.obj j) ⟶ obj (forget C) s.pt) :
            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:
simp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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)

Tactic:
have t :
      m.base =
        colimit.desc (F ⋙ PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) := by
      dsimp
      -- `colimit.hom_ext` used to be automatically applied by `ext` before https://github.com/leanprover-community/mathlib4/pull/21302
      apply colimit.hom_ext fun j => ?_
      ext
      rw [colimit.ι_desc, mapCocone_ι_app, ← w j]
      simp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
w₁ : m.base = Hom.base (desc F s)
h :
  Eq (α := s.pt.presheaf ⟶ Functor.op (Opens.map (Hom.base (desc F s))) ⋙ (colimitCocone F).pt.presheaf)
    (m.c ≫ whiskerRight (eqToHom ⋯) (colimitCocone F).pt.presheaf) (Hom.c (desc F s))
⊢ m = desc F s

Tactic:
ext : 1
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
⊢ m.base = Hom.base (desc F s)

Tactic:
t

Premises used: CategoryTheory.Limits.colimit.desc
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
⊢ m.base = Hom.base (desc F s)

Tactic:
exact t
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
⊢ Eq (α := s.pt.presheaf ⟶ Functor.op (Opens.map (Hom.base (desc F s))) ⋙ (colimitCocone F).pt.presheaf)
    (m.c ≫ whiskerRight (eqToHom ⋯) (colimitCocone F).pt.presheaf) (Hom.c (desc F s))

Tactic:
NatTrans.ext (funext fun U => limit_obj_ext fun j => ?_)

Premises used: CategoryTheory.Limits.limit_obj_ext, CategoryTheory.NatTrans.ext, funext
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
apply : colimit.cocone (F ⋙ forget C) ≅ mapCocone (forget C) (colimitCocone F)
⊢ IsColimit (F := F ⋙ forget C) (mapCocone (forget C) (colimitCocone F))

Tactic:
apply IsColimit.ofIsoColimit (colimit.isColimit _)

Premises used: CategoryTheory.Limits.colimit.isColimit, CategoryTheory.Limits.IsColimit.ofIsoColimit
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
φ : (colimit.cocone (F ⋙ forget C)).pt ≅ (colimit.cocone (F ⋙ forget C)).pt
w :
  autoParam
    (∀ (j : J),
      (colimit.cocone (F ⋙ forget C)).ι.app j ≫ Iso.hom (Iso.refl (colimit.cocone (F ⋙ forget C)).pt) =
        NatTrans.app (Cocone.ι (mapCocone (forget C) (colimitCocone F))) j)
    Cocone.ext._auto_1
⊢ colimit.cocone (F ⋙ forget C) ≅ mapCocone (forget C) (colimitCocone F)

Tactic:
fapply Cocone.ext

Premises used: CategoryTheory.Limits.Cocone.ext
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
⊢ (colimit.cocone (F ⋙ forget C)).pt ≅ (colimit.cocone (F ⋙ forget C)).pt

Tactic:
rfl
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
⊢ autoParam
    (∀ (j : J),
      (colimit.cocone (F ⋙ forget C)).ι.app j ≫ Iso.hom (Iso.refl (colimit.cocone (F ⋙ forget C)).pt) =
        NatTrans.app (Cocone.ι (mapCocone (forget C) (colimitCocone F))) j)
    Cocone.ext._auto_1

Tactic:
simp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasLimits.{v, u} C
⊢ HasColimits (PresheafedSpace.{v, u, v} C)

Tactic:
⟨fun {_ _} => ⟨fun {F} => ⟨colimitCocone F, colimitCoconeIsColimit F⟩⟩⟩

Premises used: CategoryTheory.Limits.HasColimitsOfShape.mk, Nonempty.intro, CategoryTheory.Limits.HasColimit.mk', CategoryTheory.Limits.ColimitCocone.mk, CategoryTheory.Limits.HasColimitsOfSize.mk, AlgebraicGeometry.PresheafedSpace.colimitCocone, AlgebraicGeometry.PresheafedSpace.colimitCoconeIsColimit
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
colimitCoconeIsColimit :
  obj
      (obj
        (pushforward C
          (Hom.base
            (Iso.hom
              (Iso.symm
                (colimit.isoColimitCocone
                  { cocone := colimitCocone F, isColimit := PresheafedSpace.colimitCoconeIsColimit F :
                    ColimitCocone F })))))
        { cocone := colimitCocone F, isColimit := PresheafedSpace.colimitCoconeIsColimit F :
                ColimitCocone F }.cocone.pt.presheaf)
      (op U) ≅
    limit (componentwiseDiagram F U)
⊢ (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U)

Tactic:
refine
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      ?_

Premises used: CategoryTheory.Limits.colimit.isoColimitCocone, CategoryTheory.Limits.ColimitCocone.mk, Opposite.op, CategoryTheory.Iso.trans, CategoryTheory.Iso.symm, AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso, CategoryTheory.Iso.app, AlgebraicGeometry.PresheafedSpace.colimitCoconeIsColimit
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
refine :
  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
                (Hom.base
                  (Iso.hom
                    (Iso.symm
                      (colimit.isoColimitCocone
                        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
            (op U)) :
        CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C) ≅
    componentwiseDiagram F U
⊢ obj
      (obj
        (pushforward C
          (Hom.base
            (Iso.hom
              (Iso.symm
                (colimit.isoColimitCocone
                  { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F })))))
        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }.cocone.pt.presheaf)
      (op U) ≅
    limit (componentwiseDiagram F U)

Tactic:
refine (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso ?_)

Premises used: CategoryTheory.Limits.lim, CategoryTheory.Iso.trans, CategoryTheory.Limits.limitObjIsoLimitCompEvaluation, CategoryTheory.Functor.mapIso
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
X : Jᵒᵖ
app :
  Eq (α := (Opens (↑(↑(F.obj (unop X)) : TopCat) : Type v))ᵒᵖ)
    (obj (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) (unop X))))
      (obj
        (Functor.op
          (Opens.map
            (Hom.base
              (Iso.hom
                (Iso.symm
                  (colimit.isoColimitCocone
                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
        (op U)))
    (op (obj (Opens.map (Hom.base (colimit.ι F (unop X)))) U))
⊢ obj
      (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
                  (Hom.base
                    (Iso.hom
                      (Iso.symm
                        (colimit.isoColimitCocone
                          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
              (op U)) :
          CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
      X ≅
    (componentwiseDiagram F U).obj X

Tactic:
refine (F.obj (unop X)).presheaf.mapIso (eqToIso ?_)

Premises used: CategoryTheory.eqToIso, AlgebraicGeometry.PresheafedSpace.presheaf, CategoryTheory.Functor.mapIso, Opposite.unop, CategoryTheory.Functor.obj
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
X : Jᵒᵖ
app :
  Eq (α := Set (↑((F ⋙ forget C).obj (unop X)) : Type v))
    ((fun (x : (↑((F ⋙ forget C).obj (unop X)) : Type v)) ↦
        (ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑(Limits.colimit F) : TopCat))
              (Hom.base
                (Iso.hom
                  (Iso.symm
                    (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F })))) :
            (a : (↑(↑(colimitCocone F).pt : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                  (X := (F ⋙ forget C).obj (unop X)) (Y := Limits.colimit (F ⋙ forget C))
                  (colimit.ι (F ⋙ forget C) (unop X)) :
                (a : (↑((F ⋙ forget C).obj (unop X)) : Type v)) → (↑(Limits.colimit (F ⋙ forget C)) : Type v))
              x :
            (↑(Limits.colimit (F ⋙ forget C)) : Type v))) ⁻¹'
      (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(F.obj (unop X)) : TopCat)) (Y := (↑(Limits.colimit F) : TopCat))
            (Hom.base (colimit.ι F (unop X)))) :
        (a : (↑(↑(F.obj (unop X)) : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))
⊢ Eq (α := (Opens (↑(↑(F.obj (unop X)) : TopCat) : Type v))ᵒᵖ)
    (obj (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) (unop X))))
      (obj
        (Functor.op
          (Opens.map
            (Hom.base
              (Iso.hom
                (Iso.symm
                  (colimit.isoColimitCocone
                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
        (op U)))
    (op (obj (Opens.map (Hom.base (colimit.ι F (unop X)))) U))

Tactic:
simp only [Functor.op_obj, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
      Set.preimage_preimage]

Premises used: TopologicalSpace.Opens.map_coe, Set.preimage_preimage, SetLike.ext'_iff, CategoryTheory.Functor.op_obj, Opposite.op_inj_iff
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
X : Jᵒᵖ
⊢ Eq (α := Set (↑((F ⋙ forget C).obj (unop X)) : Type v))
    ((fun (x : (↑((F ⋙ forget C).obj (unop X)) : Type v)) ↦
        (ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑(Limits.colimit F) : TopCat))
              (Hom.base
                (Iso.hom
                  (Iso.symm
                    (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F })))) :
            (a : (↑(↑(colimitCocone F).pt : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                  (X := (F ⋙ forget C).obj (unop X)) (Y := Limits.colimit (F ⋙ forget C))
                  (colimit.ι (F ⋙ forget C) (unop X)) :
                (a : (↑((F ⋙ forget C).obj (unop X)) : Type v)) → (↑(Limits.colimit (F ⋙ forget C)) : Type v))
              x :
            (↑(Limits.colimit (F ⋙ forget C)) : Type v))) ⁻¹'
      (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(F.obj (unop X)) : TopCat)) (Y := (↑(Limits.colimit F) : TopCat))
            (Hom.base (colimit.ι F (unop X)))) :
        (a : (↑(↑(F.obj (unop X)) : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))

Tactic:
congr_arg (Set.preimage · U.1) (funext fun x => ?_)

Premises used: TopologicalSpace.Opens.carrier, congr_arg, funext, Set.preimage