A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

We give a characterizations of Ramsey ultrafilters on $\mathscr P(\omega)$ in terms of functions $f:\omega^n\to\omega$ 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 $\mathcal P$.