@[simp]
theorem colimitPresheafObjIsoComponentwiseLimit_inv_ι_app (F : J ⥤ PresheafedSpace.{_, _, v} C)
    (U : Opens (Limits.colimit F).carrier) (j : J) :
    (colimitPresheafObjIsoComponentwiseLimit F U).inv ≫ (colimit.ι F j).c.app (op U) =
      limit.π _ (op j) := by
  delta colimitPresheafObjIsoComponentwiseLimit
  rw [Iso.trans_inv, Iso.trans_inv, Iso.app_inv, sheafIsoOfIso_inv, pushforwardToOfIso_app,
    congr_app (Iso.symm_inv _)]
  dsimp
  rw [map_id, comp_id, assoc, assoc, assoc, NatTrans.naturality]
  erw [← comp_c_app_assoc]
  rw [congr_app (colimit.isoColimitCocone_ι_hom _ _), assoc]
  

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
U : Opens ↑↑(Limits.colimit F)
j : J
⊢ limMap
        (NatIso.ofComponents fun X =>
            (F.obj X.unop).presheaf.mapIso
              (eqToIso
                (_ :
                  (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)))).inv ≫
      (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp
            (op
              ((Opens.map
                    (colimit.isoColimitCocone
                          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                U))).inv ≫
        ({ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.ι.app j).c.app
            (op
              ((Opens.map
                    (colimit.isoColimitCocone
                          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                U)) ≫
          (F.obj j).presheaf.map
              (eqToHom
                (_ :
                  (Opens.map
                            ({ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.ι.app
                                j).base).op.obj
                      (op
                        ((Opens.map
                              (colimit.isoColimitCocone
                                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                          U)) =
                    (Opens.map
                            (colimit.ι F j ≫
                                (colimit.isoColimitCocone
                                    { cocone := colimitCocone F,
                                      isColimit := colimitCoconeIsColimit F }).hom).base).op.obj
                      (op
                        ((Opens.map
                              (colimit.isoColimitCocone
                                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                          U)))) ≫
            ((colimit.ι F j).base _* (F.obj j).presheaf).map
              (eqToHom
                (_ :
                  op
                      {
                        carrier :=
                          ⇑(colimit.isoColimitCocone
                                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base ⁻¹'
                            (⇑(colimit.isoColimitCocone
                                      { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base ⁻¹'
                              ↑U),
                        is_open' :=
                          (_ :
                            IsOpen
                              (⇑(colimit.isoColimitCocone
                                        { cocone := colimitCocone F,
                                          isColimit := colimitCoconeIsColimit F }).hom.base ⁻¹'
                                ↑{
                                    carrier :=
                                      ⇑(colimit.isoColimitCocone
                                                { cocone := colimitCocone F,
                                                  isColimit := colimitCoconeIsColimit F }).inv.base ⁻¹'
                                        ↑U,
                                    is_open' :=
                                      (_ :
                                        IsOpen
                                          (⇑(colimit.isoColimitCocone
                                                    { cocone := colimitCocone F,
                                                      isColimit := colimitCoconeIsColimit F }).inv.base ⁻¹'
                                            ↑U)) })) } =
                    op U)) =
    limit.π (componentwiseDiagram F U) (op j)

Next tactic:
erw [limitObjIsoLimitCompEvaluation_inv_π_app_assoc, limMap_π_assoc]