The Coliseum is a location where Kairi can take a walk during her training. If she has met Hades while taking a walk in the Town Center, she will be able to fight against enemies for gold. Otherwise, she will be turned away.
After choosing an enemy to fight, the difficulty of the enemy will be compared to Kairi's Keyblade Mastery + Dark Power (Skill) + (Constitution / 2). If the difficulty is too high, she will lose, and you will lose the amount of gold for that enemy. If Kairi wins the fight, you will receive that amount of gold.
She will only start being able to fight the shadow, and defeating an enemy will unlock the next enemy to fight.