The differentiable chain functor is not homotopy equivalent to the continuous chain functor

Let S ∗ and S ∗ ∞ be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation i : S ∗ ∞ → S ∗ , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.