Mouthpiece problems................?

Alright, so my parents ordered a mouthpiece for me, specifically a Jet-Tone MF-C model. They saw Trumpet/Cornet, and umed it was trumpet. So, i have a couple of questions: Will the play be the same as a trumpet mouthpiece? And also, could I play it, or should I go ahead and buy the adapter so I can play it on my trumpet?