I haven't looked super-carefully at the assumptions in The Joy of Cats; they are described in a slightly hand-wavy way (the reader is referred to the appendix in Herrlich-Strecker, which I do not have to hand). But it's pretty clear that an assumption of ZFC plus two strong inaccessibles, one containing the other, is more than sufficient for the purposes of The Joy of Cats (they have sets contained in classes, and classes contained in "conglomerates", and they have some set-theoretical assumptions on conglomerates, the most serious of which is that a product of conglomerates indexed over a conglomerate is a conglomerate).

The formal foundations in Categories for the Working Mathematician suppose: ZFC + one inaccessible.

From the standpoint of a professional set-theorist, I think either set of assumptions would be considered fairly mild (at least when put up against large cardinal hypotheses at which a set theorist would not bat an eye), and the reaction of most people would be not to worry too much about the difference. Without having gone thoroughly through The Joy of Cats, I should think that any theorem therein that does not mention the word "conglomerate" (which might be on occasion tacit but not difficult to detect, as in "the category of categories of at most class size") would be a formal theorem under Mac Lane's declared foundations, and I am also pretty sure that Mac Lane (whom I got to know) would have no difficulty accepting ZFC + two strong inaccessibles to deal with the remainder -- it's just that he didn't need that assumption to write his book.

Without having a more specific focused question to deal with, I'm not sure one can make a more positive, guaranteed-to-be-true blanket statement about a text which is several hundred pages long.