On the endomorphisms of some sheaves of functions

Given a $C^{\mathrm{\infty }}$ real manifold $X$ and ${\mathcal{C}}_X^m$ its sheaf of $m$-times differentiable real-valued functions, we prove that the sheaf ${\mathcal{D}}_X^{m,r}$ of differential operators of order $\le m$ with coefficient functions of class $C^r$ can be obtained in terms of the sheaf $\mathcal{H}o{m}_{{\mathbb{R}}_X}({\mathcal{C}}_X^m,{\mathcal{C}}_X^r)$ of morphisms of ${\mathcal{C}}_X^m$ into ${\mathcal{C}}_X^r$. The superscripts $m$ and $r$ are integers.