Mistress Cogita says to visit the lakes to find one of the Plates.
Go to the Lakes
After you have talked to Mistress Cogita during The Researcher of Myths to begin this mission, go to the three lakes, and enter the cave in the middle of each one.
- Lake Verity is in the Obsidian Fieldlands
- Lake Valor is in the Crimson Mirelands
- Lake Acuity is in the Alabaster Icelands
Catch the Pokémon
In each cave, one of the legendary pokémon will appear. Throw a partner pokémon to begin the battle. Reduce its HP and inflict status ailments to improve your chances of capturing it, then throw Ultra Balls (or whatever types of Poké Balls you have available) until you catch it.
After you catch all three, you get the Draco Plate.