Winter Cross comments on The Internal Model Principle: A Straightforward Explanation

[Winter Cross]

Thanks for the answer! That confirms what I was thinking.

That second case: $(s_1,w_1)\to (s_2,w_2)$ and $(s_1,w_3)\to (s_2,w_2)$ surprised me since I initially thought that the IMP implied that there was an additional isomorphism between the controller and the environment. I guess that isomorphism effectively still exists since it can be created through the use of coarse graining over the controller states like you mentioned.