/-- 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 => ?_)
    

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
⊢ ((opEquiv
            ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.obj
              j₁)
            ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.obj
              j₃))
          ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                map := fun {j j'} f =>
                  (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                      (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                        (pushforwardEq
                            (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                            (F.obj j).presheaf).hom).op }.map
            (f ≫ g))).app
      U =
    ((opEquiv
            ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.obj
              j₁)
            ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.obj
              j₃))
          ({ obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.map
              f ≫
            { obj := fun j => op (colimit.ι (F ⋙ forget C) j _* (F.obj j).presheaf),
                  map := fun {j j'} f =>
                    (pushforwardMap (colimit.ι (F ⋙ forget C) j') (F.map f).c ≫
                        (Pushforward.comp (F.obj j).presheaf ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j')).inv ≫
                          (pushforwardEq
                              (_ : (F ⋙ forget C).map f ≫ colimit.ι (F ⋙ forget C) j' = colimit.ι (F ⋙ forget C) j)
                              (F.obj j).presheaf).hom).op }.map
              g)).app
      U

Next tactic:
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]