After you find the Plates that Mistress Cogita hinted at, you report back to her.

Go to the Ancient Retreat

After you complete all the Plate missions in The Researcher of Myths, go back to Jubilife Village, then talk to Ress at the Front Gate and go to the Ancient Retreat. Go into the building there and speak to Cogita.

Bring Three Pieces of Wood

Gather three pieces of wood. They can be found in the Heartwood area and Deertrack Path area of Obsidian Fieldlands, for example. When you have three, go back to Jubilife Village, then talk to Ress at the Front Gate and go to the Ancient Retreat. Go into the building there and speak to Cogita. You will get the Pixie Plate.

Go to the Celestica Ruins

Go back to Jubilife Village, then talk to Ress at the Front Gate and go to the Summit Camp in the Coronet Highlands. Go to the Celestica Ruins nearby and talk to Volo.

Prepare

The upcoming encounter is difficult. Bring plenty of Revives, Max Revives if you can get them, and other healing items. It is recommended that your party of pokémon are all level 70 or higher.

Go to the Temple of Sinnoh

Go north to the Stone Portal. Enter the cave and go through it. Go to the temple and approach Volo.

Do Battle

You must battle Volo's level 68 Spiritomb, level 68 Lucario, level 68 Garchomp, level 68 Arcanine, level 68 Togekiss, and level 68 Roserade.

Afterward, you must battle level 70 Giratina, but your pokémon will not be healed up prior to fighting it. You can't catch it, so just focus on defeating it. Giratina knows Aura Sphere (a Fighting-type move), Dragon Claw (a Dragon-type move), Earth Power (a Ground-type move), and Shadow Force (a Ghost-type move). Shadow Force has only 5 PP, so if you have a pokémon that can survive that attack, you can wait until Giratina uses Shadow Force 5 times, and then you will no longer have to worry about sending in a pokémon that is weak to Ghost-type moves.

After that, Giratina unleashes its true strength, but your pokémon will not be healed up prior to this phase. Giratina's HP and PP are restored to full. Its moves are the same as before. Again, you can't catch it, so just focus on defeating it as before.

Afterward, you get the Spooky Plate and the Azure Flute.

Return to Jubilife Village

Go to Jubilife Village and you will get the request On the Trail of Giratina.

Enter Galaxy Hall

Go into Galaxy Hall and you will get the request Incarnate Forces of Hisui.