After completing the game your trusty steed Pokémon Koraidon/Miraidon can now finally change into Battle Form and get stuck in with Pokémon battles, though they cannot be ridden in this form and also cannot be added to the PC Boxes.

What happens after you catch Koraidon in area zero?

After the ending, your Koraidon/Miraidon will be able to change to Battle Form and join the battle. But note that, they will be out of your team once you use the Legendary Ride in the field.

What happens when you catch Koraidon?

After you catch either Koraidon or Miraidon you'll need to find someone to trade with who owns Pokémon Scarlet if you own Violet, and vice versa. Most players will only be willing to trade their spare cover legendary for an equivalent of the opposite version.