Complete the indicated task to unlock the corresponding cup. Then, speak with Shinra in the Airship to enter the Colosseum.

Cup	How to unlock
Aeon Cup	Capture the eight required creatures and release them when their Fiend Tales are completed.
Almighty Shinra Cup (replaces Fiend World Cup)	Complete Omega Weapon's Fiend Tale.
Cactuar Cup	Win the Grand Cup under the Hard difficulty three times, then win it again by defeating the Cactuar team.
Chocobo Cup	Win the Grand Cup under the Hard difficulty, then win it again by defeating the Chocobo team.
Demon Cup	Capture the appointed fiends, raise all their SL, and release them.
Farplane Cup	Win all other tournaments except for the Almighty Shinra Cup.
Fiend World Cup	Capture and train a specific succession of demonic creatures.
Grand Cup	Win the Standard Cup under the Hard difficulty.
Grand Cup Hard	Win the Grand Cup three times.
Large Cup Hard	Win the Large Cup.
Large Cup	Win the Regular Har Cup.
Regular Cup Hard	Win the Regular Cup three times.
Sabotenda Cup	Win the Large Hard Cup four times.
Standard Cup Hard	Win the Standard Cup three times.
Youth League Cup	Win the Grand Hard Cup six times.