“Todd Gaither and Faith Evans are the only credited songwriters, so why would Diddy be paying Sting?”
The Police had 2 songwriters, Sting, and to a lesser extent, the drummer Stewart Copeland. They didn’t record any songs from outside songwriters.
So if someone else is credited, it’s because Sting or Copeland chose to use them as a proxy, or a pseudonym, both of which are sometimes done for various reasons in the music business.
I meant the "I'll Be Missing You" song, not Every Breath You take.
But for the Police, you are wrong, guitarist Summers wrote songs too. And Sting admitted he should have been credited as cowriter for EBYT.