Continuing about <a href="http://mathoverflow.net/questions/68009/downgrading-from-zfc-with-universes-to-zfc">this my question</a>.

Mac Lane "Categories for the Working Mathematician" and "Abstract and Concrete Categories. The Joy of Cats" use different set theory foundations.

How one to transfer theorems between these two different systems? That is if a theorem is proved in one of these two systems what can be inferred in the other?