It's a moot point, but I *really* don't understand the Doppler-shift bias mechanism either. Help?

This one of the author's other papers they cited in this one,

https://www.mdpi.com/2073-8994/15/6/1190

I'm completely lost how they're eliding between the rotation orientation of the Milky Way galaxy, and relative linear velocities with stars in other galaxies. In the special relativity argument, where does the rotation axis of the Milky Way enter?