Blue Exorcist vs. Black Rock Shooter

Rin Okumura, a student at True Cross Academy. A School where he trains to be an exorcist. Those that destroy the demons of Satan. What if he was transported into another world, but was caught up in a battle between beings he's never seen before...and one of them had blue flames like him.
Mato Kuroi, a normal high school girl. So happy to start her first day of school, she can't wait for anything except that. Her excitement, however, gets put aside as soon as she meets Okumura.
How well will Rin get along with Mato and how will he get home?

[Note: This story is based off the information given in both Anime]