/-- Given a diagram of presheafed spaces,
we can push all the presheaves forward to the colimit `X` of the underlying topological spaces,
obtaining a diagram in `(Presheaf C X)ᵒᵖ`.
-/
@[simps]
def pushforwardDiagramToColimit (F : J ⥤ PresheafedSpace.{_, _, v} C) :
    J ⥤ (Presheaf C (colimit (F ⋙ PresheafedSpace.forget C)))ᵒᵖ where
  obj j := op (colimit.ι (F ⋙ PresheafedSpace.forget C) j _* (F.obj j).presheaf)
  map {j j'} f :=
    (pushforwardMap (colimit.ι (F ⋙ PresheafedSpace.forget C) j') (F.map f).c ≫
        (Pushforward.comp (F.obj j).presheaf ((F ⋙ PresheafedSpace.forget C).map f)
              (colimit.ι (F ⋙ PresheafedSpace.forget C) j')).inv ≫
          (pushforwardEq (colimit.w (F ⋙ PresheafedSpace.forget C) f) (F.obj j).presheaf).hom).op
  map_id j := by
    apply (opEquiv _ _).injective
    refine NatTrans.ext _ _ (funext fun U => ?_)
    induction U with
    | h U =>
      rcases U with ⟨U, hU⟩
      dsimp [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op,
        pushforwardObj_obj, op_obj, Opens.map_obj, opEquiv, Equiv.coe_fn_mk, unop_comp,
        Quiver.Hom.unop_op, unop_id]
      -- Porting note : some `simp` lemmas are not picked up
      rw [NatTrans.comp_app, pushforwardMap_app, NatTrans.id_app]
      simp only [op_obj, unop_op, Opens.map_obj, map_id_c_app, Opens.map_id_obj',
        map_id, pushforwardEq_hom_app, eqToHom_op, id_eq, eqToHom_map, id_comp,
        TopCat.Presheaf.Pushforward.id_inv_app']
      rw [NatTrans.comp_app, Pushforward.comp_inv_app, id_comp]
      dsimp
      simp
  map_comp {j₁ j₂ j₃} f g := by
    apply (opEquiv _ _).injective
    refine NatTrans.ext _ _ (funext fun U => ?_)
    dsimp only [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op,
      pushforwardObj_obj, op_obj, opEquiv, Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op]
    -- Porting note : some `simp` lemmas are not picked up
    rw [NatTrans.comp_app, pushforwardMap_app, NatTrans.comp_app, Pushforward.comp_inv_app,
      id_comp, pushforwardEq_hom_app, NatTrans.comp_app, NatTrans.comp_app, NatTrans.comp_app,
      pushforwardMap_app, Pushforward.comp_inv_app, id_comp, pushforwardEq_hom_app,
      NatTrans.comp_app, NatTrans.comp_app, pushforwardEq_hom_app, Pushforward.comp_inv_app,
      id_comp, pushforwardMap_app]
    

Proof state:
case a
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
⊢ (F.map (f ≫ g)).c.app ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).op.obj U) ≫
      (F.obj j₁).presheaf.map
        (_root_.id
          (eqToHom
              (_ :
                (Opens.map (colimit.ι (F ⋙ forget C) j₁)).obj U.unop =
                  (Opens.map ((F.map (f ≫ g)).base ≫ colimit.ι (F ⋙ forget C) j₃)).obj U.unop)).op) =
    ((F.map g).c.app ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).op.obj U) ≫
        (F.obj j₂).presheaf.map
          (_root_.id
            (eqToHom
                (_ :
                  (Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj U.unop =
                    (Opens.map ((F.map g).base ≫ colimit.ι (F ⋙ forget C) j₃)).obj U.unop)).op)) ≫
      (F.map f).c.app ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).op.obj U) ≫
        (F.obj j₁).presheaf.map
          (_root_.id
            (eqToHom
                (_ :
                  (Opens.map (colimit.ι (F ⋙ forget C) j₁)).obj U.unop =
                    (Opens.map ((F.map f).base ≫ colimit.ι (F ⋙ forget C) j₂)).obj U.unop)).op)

Next tactic:
simp only [pushforwardObj_obj, op_obj, unop_op, map_comp_c_app, pushforwardMap_app,
      Opens.map_comp_obj, Pushforward.comp_inv_app, pushforwardEq_hom_app, eqToHom_op, id_eq,
      eqToHom_map, id_comp, assoc, eqToHom_trans]