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Talk to Kagetora
In Thrifty Megamart of Royal Avenue, talk to the Veteran cashier and he will tell you about some trainers who use Eeveelutions that he wants you to meet.
Find the Trainers
Go to the third floor of the Trainers' School on Melemele Island and talk to the woman to the right of the stairs. Vaporeon User Polly has a level 55 Vaporeon. You get 11000 Pokédollars for winning.
Go to Tide Song Hotel, north of Heahea City on Akala Island. Go to the left of the front desk in the lobby and talk to the janitor. Flareon User Chad has a level 55 Flareon. You get 2200 Pokédollars for winning.
Go to the Community Center in Malie City on Ula'ula Island. Talk to the woman sitting near the TV in the lobby. Jolteon User Jane has a level 55 Jolteon. You get 11000 Pokédollars for winning.
Go to the Geothermal Plant northwest of Route 12 on Ula'ula Island and talk to the janitor who is watching TV. Espeon User Ishaan has a level 55 Espeon. You get 2200 Pokédollars for winning.
Go to Hau'oli Cemetery on Melemele Island and talk to the guy in the northwest corner. Umbreon User Braiden has a level 55 Umbreon. You get 11000 Pokédollars for winning.
Go to Iki Town on Melemele Island and talk to the old woman in the southwest house. Glaceon User Rea has a level 55 Glaceon. You get 11000 Pokédollars for winning.
Go to the far north end of Hano Beach and talk to the woman there. Leafeon User Linnea has a level 55 Leafeon. You get 1320 Pokédollars for winning.
Go into the northeast house of Seafolk Village and talk to the preschooler in the right side of the house. This is Sakura's granddaughter. Sylveon User Kira has a level 54 Sylveon. You get 648 Pokédollars for winning.
Get your Reward
Go back to Thrifty Megamart in Royal Avenue on Akala Island and talk to Kagetora again. He will ask to battle you. Eevee User Kagetora has a level 57 Eevee. You get 5016 Pokédollars for winning. Afterward he gives you the Z-Crystal, Eevium Z, which upgrades Eevee's move Last Resort.