Speak with Shinra in the Airship to enter the Colosseum.

Unlockable	How to unlock
Aeon Cup	Capture the appointed fiends and raise all them to become the Aeon.
Chocobo Cup	Win Large Hard Cup twice.
Demon Cup	Capture the appointed fiends and raise all their SL and release them.
Farplane Cup	Win all other Cups.
Large Cup	Win Regular Har Cup once.
Large Cup Hard	Win Large Cup once.
Regular Cuo Hard	Win Regular Cup 3 times.
Sabotenda Cup	Win Large Hard Cup 4 times.
Youth League Cup	Win Large Hard Cup 6 times.