A representation of sup-completion

It was showed by Donner in 1982 that every order complete vector lattice $X$ may be embedded into a cone $X^s$, called the sup-completion of $X$. We show that if one represents the universal completion of $X$ as $C^{\mathrm{\infty }}(K)$, then $X^s$ is the set of all continuous functions from $K$ to $[-\mathrm{\infty },\mathrm{\infty }]$ that dominate some element of $X$. This provides a functional representation of $X^s$, as well as an easy alternative proof of its existence.