On the canonical Ramsey theorem of Erdös and Rado and Ramsey ultrafilters

We give a characterizations of Ramsey ultrafilters on $\omega$ in terms of functions $f\colon\omega^n\to\omega$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal{P}$ of $[\omega]^n$ there is a finite partition $\mathcal{Q}$ of $[\omega]^{2n}$ such that any set $X\subseteq\omega$ that is homogeneous for $\mathcal{Q}$ is a finite union of sets that are canonical for $\mathcal{P}$.