I have Ultra Moon, and my brother has Ultra Sun. I have the cosmog that you can get from the mirror world. If I transfer the cosmog into Ultra Sun, it will evolve into a cosmoem, but will it it be lunala or solgaleo after that?
Its evolution is determined by the game it's in when it evolves. If Cosmoem evolves in Ultra Sun, you're getting a Solgaleo regardless of the game from which it spawned as a Cosmog.
Source: DB
