Adrian Mathias has written some excellent articles comparing the specific set theory used by Mac Lane and used in other parts of category theory.
His article The strength of Mac Lane set theory is a detailed analysis of the strength of Mac Lane's set theory in comparison with other theories.
Adrian Mathias, "What is Mac Lane missing?" (a comment on the foundational stance of Saunders Mac Lane; MR 94g:03010; published with a reply by Mac Lane, MR 94g:03011, in Set Theory of the Continuum, ed. H. Judah, W.Just, H.Woodin; Mathematical Sciences Research Institute Publications Volume 26, Springer-Verlag, 1992.)
See also Adrian Mathias, "Strong statements of analysis."
Further papers available on Mathias's web page depository.