Edgeworth isn't the type to nominate anyone his rival for sometime now, he has grown out of it, or it's at least what he tells himself, it's unspoken - you are indeed his rival. Not on a personal level but almost there.

Edgeworth’s world doesn't revolve around Phoenix's existence, and the only other lawyer that has been capable of defeating him non-stop is no one else than you, it's childish rivalry, he knows, but he can't help a frown when you walk by him after you claim another victory.

“So, what is it that you wish to speak with me? Have you come to laugh at me?” The prosecutor grunts, though, he notices how unreasonable he's being, taking a deep breath in. “...Just state your business.” Edgeworth requests, not wanting to lose his composure around you.