Tyson will occasionally receive challenges issued by Moriarty with wax seals and responds with his own seals when replying or returning challenges.
He may offer suggestions or ask if one of the Moriarty Mirrors is capable of something in a way that is intended to prompt a response, he often even challenges what the Moriarty Mirrors say and do, but the only time Tyson throws down the gauntlet and issues a challenge is through those wax-sealed letters.
Tyson going out of the way to play into the Moriarty Mirrors theatrics means that he is daring them to do their worst and prove why they are the rulers of a criminal empire. It is also one the rare times that Tyson will make a request of them.