This is a challange between Naruto and Boku no hero acadamia where Naruto was born in the BnHA world.
He has 2 quirks. The first one is called superhuman. This quirk allows him to basically be a superhuman ( for example he has super strength, super speed, advanced reflexes and healing, as well as advanced senses( sight, smell etc.)
His second quirk is called information. This quirk allows him to acquire all the information of anything or anyone he touches. For instance if he touches someone he will know every single thing about them and if he so wishes he can choose to forget or remember any info he gets. He will remember forgeting the info but not who or what the info was about.
The story starts after the first attack by The League of Villains. The teachers at UA are holding a meeting when Naruto shows up. He greets them all by shaking their hands. He tells them that he is the vigilante that has been going around and helping the people and he explains to them about his quirks ( specifically information ). This is when the teachers realise that he shook all of their hands. They are about to attack him when he quickly explains to them that he is there to help. After talking for awhile the teachers agree that he needs to stay close so that is how he ends up teaching class 1-A alongside eraserhead. He also tells the class about his quirks. And explains to Deku that he knows about One for All.
They also make him an official hero because of his good work as a vigilante.
Naruto x Midnight or Momo Yaoyorozu
After that you can pretty much do what you want so long as Naruto stays on the hero side.