-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
theorem desc_c_naturality (F : J ⥤ PresheafedSpace.{_, _, v} C) (s : Cocone F)
    {U V : (Opens s.pt.carrier)ᵒᵖ} (i : U ⟶ V) :
    s.pt.presheaf.map i ≫ descCApp F s V =
      descCApp F s U ≫
        (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s) _* (colimitCocone F).pt.presheaf).map
          i

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
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
⊢ s.pt.presheaf.map i ≫
      limit.lift
          ((pushforwardDiagramToColimit F).leftOp ⋙
            (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))))
          { pt := s.pt.presheaf.obj V,
            π :=
              { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                naturality := ⋯ } } ≫
        (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
            (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv =
    (limit.lift
          ((pushforwardDiagramToColimit F).leftOp ⋙
            (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U))))
          { pt := s.pt.presheaf.obj U,
            π :=
              { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                naturality := ⋯ } } ≫
        (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
            (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫
      (limit (pushforwardDiagramToColimit F).leftOp).map
        ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op

Tactic:
refine limit_obj_ext (fun j => ?_)

Result:
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
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
j : Jᵒᵖ
⊢ (s.pt.presheaf.map i ≫
        limit.lift
            ((pushforwardDiagramToColimit F).leftOp ⋙
              (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))))
            { pt := s.pt.presheaf.obj V,
              π :=
                { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ } } ≫
          (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv) ≫
      (limit.π (pushforwardDiagramToColimit F).leftOp j).app
        (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))) =
    ((limit.lift
            ((pushforwardDiagramToColimit F).leftOp ⋙
              (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U))))
            { pt := s.pt.presheaf.obj U,
              π :=
                { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ } } ≫
          (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫
        (limit (pushforwardDiagramToColimit F).leftOp).map
          ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op) ≫
      (limit.π (pushforwardDiagramToColimit F).leftOp j).app
        (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))

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
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
j : Jᵒᵖ
w :
  (Opens.map (colimit.ι (F ⋙ forget C) (unop j))).map
      ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop) =
    eqToHom ⋯ ≫ (Opens.map (((forget C).mapCocone s).ι.app (unop j))).map i.unop ≫ eqToHom ⋯
⊢ (s.pt.presheaf.map i ≫
        limit.lift
            ((pushforwardDiagramToColimit F).leftOp ⋙
              (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))))
            { pt := s.pt.presheaf.obj V,
              π :=
                { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ } } ≫
          (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv) ≫
      (limit.π (pushforwardDiagramToColimit F).leftOp j).app
        (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))) =
    ((limit.lift
            ((pushforwardDiagramToColimit F).leftOp ⋙
              (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj
                (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U))))
            { pt := s.pt.presheaf.obj U,
              π :=
                { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                  naturality := ⋯ } } ≫
          (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
              (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫
        (limit (pushforwardDiagramToColimit F).leftOp).map
          ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op) ≫
      (limit.π (pushforwardDiagramToColimit F).leftOp j).app
        (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))

Tactic:
simp [congr_arg Quiver.Hom.op w]

Result:
no goals