Thee forms of Autonomy

I am reading Paul Guyer's Kant.  Guyer mentions how in the Critique of the Power of Judgment, Kant says that "each of three main cognitive powers - theoretical understanding practical reason, and the power of judgment - can be understood as a form of authonomy" (Guyer's words, p 5; cf. V:196)

I was wondering if each of these powers somehow coincided with the three "authorities" in the state, the legislative authority, the executive authority, and the judicial authority. Or whether the three critiques coincided with the three definitive articles of perpetual peace. Interestingly, we might see cosmopolitan right as about judgment. In the third critique, Kant says that by looking at nature's organization, we can (according to Guyer) see that "nature is hospitable [!] to our most general cognitive and practical objectives." (Guyer, p. 6) I wonder if this maps on to the right to hospitality, in that cosmopolitan right means the ordering of the earth so it is hospitable our cognitive and practical objectives (such as exchanging ideas and goods). Of course, I can't read too much into Guyer's words, but it does make me wonder what is meant by hospitality (and what would be required to make the globe hospitable for our objectives).