So, you are about to catch the lake trinity, also known as the "Spirits of the Lake". It is a creature of knowledge Wuxi, a creature of emotion Mesprit, and a creature of willpower Azelf. Capturing them requires skill, patience and time.
Steps

Step 1. Before you start, you must defeat the Galaxies team and stop their plan to destroy the world
You will also need to collect all eight hall icons. But if you want, you can get the last badge before trying to catch the trinity from the lake. You can even go through the Pokémon League before moving on to catch this trio. It's all up to you. You will also need a Pokémon with the Surf skill.

Step 2. Travel to one of the three Sinnoh lakes
On Route 214 is Lake Valor, Lake of Insight is near Snowpoint, and Lake of Truth is near Twinleaf (your character's home).

Step 3. When you swim on the lake, in the center you will find a small island with a cave in the middle, disembark on it

Step 4. Enter the cave on the island
Depending on the lake, you will find Azelf, Uxi or Mesprit in the cave (Azelf on Lake Valor, Uxi on Lake Acute, and Msprit on the Isle of Truth).

Step 5. Press the "A" button in front of the Pokemon to talk to him and initiate a battle

Step 6. If you are fighting Azelf or Wuxi, weaken them to a significant level, and then (if you like) impose some status on them (Sleep or Paralysis is best)

Step 7. Throw duskballs at the Pokémon (since you are in a cave, they will be very effective), ultraballs (they are always very good) or timerballs (if the battle lasts longer than 30 or 40 turns)
You can of course not use these balls if you like, but their use is recommended. If you lower the health of a Pokemon to 1, you can catch it with regular Pokeballs, it's not that difficult.

Step 8. It is recommended to catch Mesprit either before catching Uksi and Azelf, or after them
When you go to Mesprit to talk to him, you will see his picture (it will be added to your Pokedex). After that, he will run away. Professor Rowan will come to you and tell you that Mesprit wants to play catch-up with you.

Step 9. You can use the Map Marker Poketch application to see where Mesprit is currently located
But every time you switch from one path to another or try to fly to Mesprit, it will change its location (read "Tips").

Step 10. When you try to catch Mesprit, use the Mean Look skill (although you will need a very fast Pokémon to do this, for example, Globat) so that he does not run away every time you try to hit or catch him
When you loosen it enough and meet again, it is recommended to use quickballs.

Step 11. Enjoy having Trinity Lake
Advice
- Remember to save before fighting them, in case you accidentally knock them out.
- Good advice for catching Mesprit easily: go where one path ends and another begins, and stand near the tall grass. Keep walking between the tracks and Mesprit will keep changing its location. Continue walking like this until Mesprit is in the grass near you. Then approach him and join the battle.
- Bring a lot of Ultra Balls or Duskballs with you if you catch them at night, as this Trio has a catch rate of 3 like other Legendary Pokémon.
- A cricket with Sing, False Swipe (TM54) and X-Scissor for initial damage will help you a lot. With it, you can catch almost all Pokémon.
- If Azelf knocks you out before you catch him, take Mr. Maim with you with Encore / Light Screen skills or Goldak with Amnesia to defend against Azelf's attacks. You can even take Wuxi if you have already caught her.
- Status effects like freezing, paralysis and others work very well for these Pokémon.
- Take a level 1 Pokémon with you, preferably Ratata with Focus Sash and Endeavor, and let the enemy hit her. Thanks to the Focus Sash skill, Ratata only has 1 health remaining. After that, use Endeavor to bring the enemy's health to 1.
- To catch the trinity, use Pokémon level 50 and above, since this is the level of Azelf, Mesprit and Uksi.
- Fun Fact: During the battle with Mesprite, instead of the Wild Pokémon theme song, the Trinity Lake theme music will play in the background.
Warnings
- Don't be discouraged when looking for Mesprit or when trying to catch Azelf or Uxi. If you start to get very nervous about failures, take a break and rest for a couple of minutes.
- See, don't knock them out. It is quite difficult to catch them, but if you save yourself before the battle, you can try again.