User blog comment:Staw-Hat Luffy/Why Kaido?/@comment-4889755-20130204233830/@comment-4605926-20130205161431

Well, it could be a reason. However, I have a different opinion on the matter (which doesn't negate yours of course).

I think Law is involved with Doflamingo (which is affiliated with Kaido) more than he would've liked to be or more than he can handle at the moment. Therefore he is trying to sabotage Kaido's battle power (which partially consists of artificial devil fruit users) and with that Doflamingo's influence and what not since Doflamingo will be put to blame (for some of the results) as stated here.

Moreover, Law never said he wanted to take down all four Yonko, it was Luffy who did. Law's sole intention (of which we are currently aware of) is to take out Kaido and Doflamingo along with him (or at least a significant proportion of their battle power and influence). After that, the fate of the alliance only time will reveal.