where wanting to know the truth about the D. is not an ambition? if so robin doesn't have an ambition. It's okay for a character to have 2 dreams, Luffy himself does. knowing the truth about the D. is law's new ambition, it's literally written in chapter 996
calm down there. having CoC because of name makes no sense. if that were the case, all you had to do was name everyone king and everyone had CoC. if that were the case every king would have CoC, like king riku, king elizabelo or cobra.
if king can should have CoC because he wanted to make kaido PK then law should have too since he wants to know the truth about the D. becoming PK and discovering the truth about the D. are equally lofty goals as they both involve finding the 4 roads poneglyphs and go to laughtale, which means to challenge the yonkous and the world government