-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
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
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
j : J
⊢ Eq (α :=
    limit (componentwiseDiagram F U) ⟶ obj (obj (pushforward C (Hom.base (colimit.ι F j))) (F.obj j).presheaf) (op U))
    (Iso.inv (colimitPresheafObjIsoComponentwiseLimit F U) ≫ NatTrans.app (Hom.c (colimit.ι F j)) (op U) :
      limit (componentwiseDiagram F U) ⟶ obj (obj (pushforward C (Hom.base (colimit.ι F j))) (F.obj j).presheaf) (op U))
    (limit.π (componentwiseDiagram F U) (op j) :
      limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
have delta : Eq (α :=
  limit (componentwiseDiagram F U) ⟶ obj (obj (pushforward C (Hom.base (colimit.ι F j))) (F.obj j).presheaf) (op U))
  (Iso.inv
      (Iso.trans (Y :=
        obj
          (obj
            (pushforward C
              (Hom.base
                (Iso.hom
                  (Iso.symm
                    (colimit.isoColimitCocone
                      { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F })))))
            { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }.cocone.pt.presheaf)
          (op U))
        (Iso.app
          (sheafIsoOfIso
            (Iso.symm
              (colimit.isoColimitCocone
                { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F })))
          (op U))
        (Iso.trans (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)
                  (obj
                    (Functor.op
                      (Opens.map
                        (Hom.base
                          (Iso.hom
                            (Iso.symm
                              (colimit.isoColimitCocone
                                { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F :
                                  ColimitCocone F }))))))
                    (op U)) :
                CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
          (limitObjIsoLimitCompEvaluation (Functor.leftOp (pushforwardDiagramToColimit F))
              (obj
                (Functor.op
                  (Opens.map
                    (Hom.base
                      (Iso.hom
                        (Iso.symm
                          (colimit.isoColimitCocone
                            { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
                (op U)) :
            obj
                (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)
                (obj
                  (Functor.op
                    (Opens.map
                      (Hom.base
                        (Iso.hom
                          (Iso.symm
                            (colimit.isoColimitCocone
                              { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F :
                                ColimitCocone F }))))))
                  (op 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)
                      (obj
                        (Functor.op
                          (Opens.map
                            (Hom.base
                              (Iso.hom
                                (Iso.symm
                                  (colimit.isoColimitCocone
                                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F :
                                      ColimitCocone F }))))))
                        (op U)) :
                    CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)))
          (mapIso lim (NatIso.ofComponents (fun (X : Jᵒᵖ) ↦ mapIso (F.obj (unop X)).presheaf (eqToIso sorry)) sorry) :
            obj (C := CategoryTheory.Functor.{v', v, u', u} Jᵒᵖ C) lim
                (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)
                      (obj
                        (Functor.op
                          (Opens.map
                            (Hom.base
                              (Iso.hom
                                (Iso.symm
                                  (colimit.isoColimitCocone
                                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F :
                                      ColimitCocone F }))))))
                        (op U)) :
                    CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C)) ≅
              obj lim (componentwiseDiagram F U)))) ≫
    NatTrans.app (Hom.c (colimit.ι F j)) (op U))
  (limit.π (componentwiseDiagram F U) (op j))
Tactic: delta colimitPresheafObjIsoComponentwiseLimit