Complete the Fiend Tales for the following monsters. Capture them, raise their experience by five levels, and release them in the Creature History menu to complete their Fiend Tales. They will then evolve into Aeons. After all of the monsters have evolved, the Aeon cup will be unlocked.

Cup	How to unlock
Chapter 1-4 (Besaid Island)	Large trap pod - Flame Dragon (Ifrit)
Chapter 2 (Moonflow, Bevelle, Calm Lands)	Small trap pod - Flan Blanco (Shiva)
Chapter 3-5 (Mi'ihen Highroad, Zanarkand Ruins)	Small trap pod - Mycotoxin (Valefor)
Chapter 5 (Bevelle)	Medium trap pod - Critical Bug (Yojimbo)
Chapter 5 (Bevelle)	Large trap pod - Azi Dahaka (Bahamut)
Chapter 5 (Bikanael Desert - after completing mission)	Large trap pod - Jumbo Cactuar (Magus Sisters)
Chapter 5 (Bevelle - after clearing at least Cloister level 20)	Large trap pod - Aranea (Anima)
Chapter 5 (Thunder Plains - after clearing Thunder Plains )	Large trap pod - Machina Striker (Ixion)