-- 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
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))
    (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))) 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))) U)) :
              CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
          { pt := s.pt.presheaf.obj (op U),
            π :=
              {
                app := fun (j : Jᵒᵖ) ↦
                  NatTrans.app (Hom.c (s.ι.app (unop j))) (op U) ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯),
                naturality := ⋯ :
                NatTrans (obj (Functor.const Jᵒᵖ) (s.pt.presheaf.obj (op 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))) U)) :
                      CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤
                        C)) } })
        (limit.π
            (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))) U)) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
            (op j) ≫
          NatTrans.app (whiskerRight (eqToHom ⋯) (F.obj j).presheaf) (op U)) :
      s.pt.presheaf.obj (op U) ⟶ ((Opens.map (s.ι.app j).base).op ⋙ (F.obj j).presheaf).obj (op U))
    ((s.ι.app j).c.app (op U) :
      s.pt.presheaf.obj (op U) ⟶ ((Opens.map (s.ι.app j).base).op ⋙ (F.obj j).presheaf).obj (op U))

Tactic:
simp