If you see the game in that section, then it’s synced and there’s nothing more you need to do. If you’re facing the error upon attempting to start a game that states to verify game files with Steam, that doesn’t refer to ownership verification, it refers to file integrity verification (i.e. repairing the game). In that case, check my comment from here: https://forum.plutonium.pw/topic/39666/error-during-initialization-invalid-patch-fastfiles-please-verify-your-game-files-with-steam/2