Let $\mathrm{Man}$ be the category of smooth manifolds (2nd countable, Hausdorff, no boundary, not necessarily compact) and smooth maps, and let $M$ be an object thereof. Is the presheaf $S \mapsto C^\infty(S \times M,\mathbb{R})$ on $\mathrm{Man}$ isomorphic to a filtered direct limit of representable functors?

("Yes" when $\dim(M) = 0$, possible strategy in general: If $C^\infty(M,\mathbb{R})$ is made into some kind of infinite-dimensional manifold, perhaps the finite-dimensional submanifolds of it would form a filtered category.)