How to transform an Ubuntu Unity to Ubuntu Mate?

The way you word this question the only way to do this is by re-installing from an Ubuntu Mate ISO are formatting your current installation.

By completly removing Unity you will break your system. By 1st installing Mate and then removing Unity you might break your system. Here is a method for removing Unity where someone wanted Cinnamon installed. That will work for Mate too if change the personal archive to this.

But if those commands are not up to date you will leave traces of Unity on your system. So, to me, a re-install is the only method to meet all your demands.

Ubunutu mate provides nice guide to replace Unity with Mate desktop. http://wiki.mate-desktop.org/replace_unity_by_mate