-- 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
U V : (Opens (↑(↑s.pt : TopCat) : Type v))ᵒᵖ
i : U ⟶ V
⊢ Eq (α :=
    s.pt.presheaf.obj U ⟶
      obj (C := (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)
        (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)))
        (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V)) :
          (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ))
    (s.pt.presheaf.map i ≫
      CategoryStruct.comp (Y :=
        limit
          (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)
                (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
        (limit.lift
            (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)
                  (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
            { pt := s.pt.presheaf.obj V,
              π :=
                {
                  app := fun (j : Jᵒᵖ) ↦
                    NatTrans.app (Hom.c (s.ι.app (unop j))) V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ :
                  NatTrans (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj V))
                    (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)
                          (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                        CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                          C)) } } :
          Cone.pt (F :=
              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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
              { pt := s.pt.presheaf.obj V,
                π :=
                  {
                    app := fun (j : Jᵒᵖ) ↦
                      NatTrans.app (Hom.c (s.ι.app (unop j))) V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                    naturality := ⋯ :
                    NatTrans (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj V))
                      (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)
                            (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                          CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                            C)) } :
                Cone
                  (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)
                        (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                      CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                        C)) } ⟶
            limit
              (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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
        (Iso.inv
            (limitObjIsoLimitCompEvaluation (Functor.leftOp (pushforwardDiagramToColimit F))
              (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V)))) :
          limit
              (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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)) ⟶
            obj (C := (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)
              (limit
                  (Functor.leftOp (pushforwardDiagramToColimit F) :
                    Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
              (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop V)) :
                (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)))
    (CategoryStruct.comp (Y :=
      obj (C := (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)
        (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)))
        (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)) :
          (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ))
      (CategoryStruct.comp (Y :=
        limit
          (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)
                (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
        (limit.lift
            (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)
                  (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
            { pt := s.pt.presheaf.obj U,
              π :=
                {
                  app := fun (j : Jᵒᵖ) ↦
                    NatTrans.app (Hom.c (s.ι.app (unop j))) U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ :
                  NatTrans (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U))
                    (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)
                          (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                        CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                          C)) } } :
          Cone.pt (F :=
              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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
              { pt := s.pt.presheaf.obj U,
                π :=
                  {
                    app := fun (j : Jᵒᵖ) ↦
                      NatTrans.app (Hom.c (s.ι.app (unop j))) U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                    naturality := ⋯ :
                    NatTrans (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj U))
                      (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)
                            (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                          CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                            C)) } :
                Cone
                  (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)
                        (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                      CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                        C)) } ⟶
            limit
              (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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
        (Iso.inv
            (limitObjIsoLimitCompEvaluation (Functor.leftOp (pushforwardDiagramToColimit F))
              (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)))) :
          limit
              (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)
                    (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U))) :
                  CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)) ⟶
            obj (C := (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)
              (limit
                  (Functor.leftOp (pushforwardDiagramToColimit F) :
                    Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
              (op (obj (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) (unop U)) :
                (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ)))
      (map
        (limit
          (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C))))
        (Quiver.Hom.op (map (Opens.map (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s))) i.unop))))

Tactic:
limit_obj_ext (fun j => ?_)

Premises used: CategoryTheory.Limits.limit_obj_ext, TopCat.Presheaf