A note on ultrahomogeneous unary algebras

In a recent paper \cite{1}, Quinn-Gregson fully classified ultrahomogeneous mono-unary algebras. In particular, he proved that every locally finite mono-unary algebra with finitely many 1-orbits is \omega-categorical, and every 1-ultrahomogeneous mono-unary algebra is ultrahomogeneous. He then asked if these two results can be generalized to unary algebras. This short note answers the second question negatively by providing a simple counterexample. We also show that the first question has a positive answer in ``tree-like" cases which covers his result to mono-unary algebras and has a small combinatorial implication.