The "note to the reader" clarifies things a bit -- he's aiming for requiring "no more mathematical knowledge than might be acquired from an undergraduate degree at an ordinary British university". Though he does not specify whether he has in mind a mathematics degree, I think this can be deduced from the fact that he indicates that the text developed out of a master's-level course.
The text developed from approximately six lectures' worth of the MMath-level 24-lecture Part III Introduction to Category Theory at Cambridge, I believe. (Source: I took that course last year, and was part of a small reading group studying Leinster's Basic Category Theory at that time. We found that book really, really helpful.)
