Why did Caesar let Law in Punk Hazard?
The reasons Law told Ceasar:


  I'm looking into research that the government is carrying out. This facility should contain valuable traces of research that is still going on to this day.


(chapter 666, page 5)

Ceasar's exact reaction was:


  I don't see how I can trust you, but I suppose your proposition can't hurt


(chapter 666, page 6)

So Law's story checked out, he didn't seem like a threat, and even though he wasn't trusted, Ceasar thought he would benefit from keeping Law around.