@[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
  

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) ≫
        ((colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c.app
              (op
                ((Opens.map
                      (colimit.isoColimitCocone
                            { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                  U)) ≫
            (Limits.colimit F).presheaf.map
              (𝟙
                (op
                  ((Opens.map
                        (colimit.isoColimitCocone
                              { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).obj
                    ((Opens.map
                          (colimit.isoColimitCocone
                                { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj
                      U))))) ≫
          (Limits.colimit F).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))) ≫
      (colimit.ι F j).c.app (op U) =
    limit.π (componentwiseDiagram F U) (op j)

Next tactic:
rw [map_id, comp_id, assoc, assoc, assoc, NatTrans.naturality]