So after Rose let go of Jack he sank into the ocean and then washed up on the shore in the first scene of Inception. Between the heartbreak from Rose and the madness he went through due to Inception, he was never the same. He couldn’t tell the difference between reality and dreams anymore so they took him to Shutter Island due to his madness. But he failed to accept that he was insane so they convinced him he was a cop and that they needed him to solve a case that never even happened in attempt to cure him. But it failed so they continue the cycle until they cure him.