Lillie’s Cosmog features quite heavily in the story, but you can catch one for your own after you beat the Elite Four. This adorable ethereal Pokémon evolves into Cosmoem at level 43 before transforming into Solgaleo or Lunala (depending on your version of the game) at level 53, so you’ll definitely want to snag one if you’re looking to complete the evolutionary set.
Once you’ve cemented your title as Champion, head back to the Altar of Sunne or Moone and hop through the portal you find there. If you’re playing Pokémon Moon, the portal will only be open during the day, and vice versa in Pokémon Sun. Make your way to the Lake of Sunne or Moone and climb to the top of the platform, where you’ll find a little Cosmog for your very own. A cutscene will play and you’ll then be given the option to add it to your party or Box.
While you can essentially waltz right up to Cosmog and add it to your party, the ominous looking Necrozma puts up more of a fight. This pure Psychic Pokémon appears at level 75, so you had best come prepared for a challenge.
Return to Ten Carat Hill on Melemele Island and make your way to the Farthest Hollow section of the map. Bring a Max Repel to ward off any weaker Pokémon and then roam the wild grass until Necrozma appears. After that, it’s simply a matter of surviving its attacks and wittling down its HP until you can nab it.
You hear about the Tapus throughout your journey across Alola and now at long last you can finally catch them and add them to your team. There are four in all, so stock up on those Pokéballs. These island deities won’t be caught without a fight.
This is the first of the Tapus you encounter in the early moments of the game, and it’s also the easiest to find. Once you’re dubbed Champion, you’ll travel with Lillie to Tapu Koko’s shrine on Melemele Island, where the Pokémon will then challenge you.
Tapu Lele guards Akala Island and lurks within the Ruins of Life. You’ll need the Machamp PokéRide to get far enough in the ruins to challenge Tapu Lele, but it’s otherwise fairly straightforward.
Hop on over to Ula’ula Island next to capture Tapu Bulu. This Grass/Fairy-type guardian hides in the Ruins of Abundance, but you can only access his lair by traveling through the island’s sweeping desert first. Make your way to Haina Desert and travel Up, Right, Up, Right, Left, Down, Up to find it.
The last deity, Tapu Fini, dwells on Poni Island in the Ruins of Hope. The Water/Fairy-type Pokémon, again, can only be approached with the aid of the Machamp PokéRide.
These legendaries make excellent additions to your team, but they also present a fun end-game challenge for completionists. Take your time, be sure to save your game before any encounters, and enjoy exploring Alola as Pokémon Champion.