Thoughts and beliefs are system II stuff. If you do X because you believe Y, that's system II.

Intuition is often when you do things and have no reason for reasons that can't be expressed in language as such it's a lot harder to investigate. The kind of intuition that lets a fireman feel fear when he can't see any logical reason and then quit the building to safe his life, can't be broken down analytically.

I don't see a good reason to speak in terms of system I and system II when speaking about sitting down to retrospect.