When you unlock the Coliseum, you will get a chance to fight against other players in the game for a chance to be in the top division.
You assemble 15 Heroes to battle teams assembled by other players. Setting your defensive teams creates a snapshot of those teams. This allows you to use different Friend Disks when defending than when attacking.
You have to be victorious in all 3 battles to win and advance! The Coliseum has Copper, Bronze, Silver, Gold, and Platinum Tiers that rank players in the game. Each contain five Divisions.
The Challenger Tier contains the best players in the game. You'll be promoted here from Platinum.