I think in the instruction manual for the first game it says the actual trial system is called turnabout something or another.
Anyway, I'm halfway through the final case in the game, and I'm definitly not liking this game as much as the previous 3. I like the characters just fine, but they just don't click as well as the old gang did. I think its because I'm not as interested in the overall story of Apollo and Trucy like I was for Edgeworth/Phoenix/Maya in the first game, Fran/Edgeworth/Pearl/Maya in the second, and Phoenix/Mia/Maya/Godot in the last game.
I think another part of it is that Phoenix was the character through which we saw the world and he had a definite connection to most of those characters. He cared about them and almost by proxy, we cared about them. The same effect isn't there for Apollo. Other than Trucy, who does he have a relationship that he seems to care about? Phoenix supposedly plays the mentor role, but they don't have nearly as good of a relationship or back and forth that Phoenix had with Gumshoe and Edgeworth and even Fran.
Overall, I like the new gameplay mechanics added in, but I think the writers either need to make this story told through Phoenix's eyes and bring back the Feys' and Edgeworth, or they need to have something happen to Phoenix so that Apollo can develop relationships with the characters of Ace Attorney 5.
My final example is going back and playing 1-2. The case is relatively easy, but because of who was killed, I get immense satisfaction from nailing the criminal. There's about 4 or 5 characters in the prequels I feel the same way about. So far, the only one I'd care about if they offed is Trucy, and that'll never happen. That just shows that the connection hasn't been made with the new cast yet.