-- 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
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
j : J
rw :
  Eq (α :=
    limit (componentwiseDiagram F U) ⟶ obj (obj (pushforward C (Hom.base (colimit.ι F j))) (F.obj j).presheaf) (op U))
    (CategoryStruct.comp (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))
          (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)
                    (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)))
            (Iso.inv
                (mapIso lim (NatIso.ofComponents (fun (X : Jᵒᵖ) ↦ mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯)) :
              limit (componentwiseDiagram F 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)))
            (Iso.inv
                (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))) :
              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)) ⟶
                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)))
          (CategoryStruct.comp (Y :=
            obj (C :=
              (Opens
                  (↑(obj (forget C)
                        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F :
                              ColimitCocone F }.cocone.pt) :
                    Type v))ᵒᵖ)
              (obj
                (pushforward C
                  (Iso.hom
                    (Iso.symm
                      (mapIso (forget C)
                        (Iso.symm
                          (colimit.isoColimitCocone
                            { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }))))))
                (Limits.colimit F).presheaf)
              (op
                (obj
                  (Opens.map
                    (Iso.inv
                      (Iso.symm
                        (mapIso (forget C)
                          (Iso.symm
                            (colimit.isoColimitCocone
                              { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }))))))
                  (unop (op U)))))
            (NatTrans.app
                (Hom.c
                  (Iso.hom
                    (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F })))
                (op
                  (obj
                    (Opens.map
                      (Iso.inv
                        (Iso.symm
                          (mapIso (forget C)
                            (Iso.symm
                              (colimit.isoColimitCocone
                                { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }))))))
                    (unop (op U)))) ≫
              (Limits.colimit F).presheaf.map (eqToHom ⋯))
            ((Limits.colimit F).presheaf.map (eqToHom ⋯))) ≫
        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))
⊢ 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 ⋯)) ⋯) :
              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:
rw [Iso.trans_inv, Iso.trans_inv, Iso.app_inv, sheafIsoOfIso_inv, pushforwardToOfIso_app,
    congr_app (Iso.symm_inv _)]

Premises used: CategoryTheory.Iso.app_inv, TopCat.Presheaf.pushforwardToOfIso_app, CategoryTheory.Iso.symm_inv, AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso_inv, AlgebraicGeometry.PresheafedSpace.congr_app, CategoryTheory.Iso.trans_inv