It's pretty heavy going, but with considerable aid from Wikipedia I'm about a quarter of the way through without being completely lost (I think). This is pretty good going for a mathematics text for me.
Taylor's attitude is interesting:
Instead of trying to find one mathematical world (set of axioms) in which all the mathematics we are interested in is provable, he asks what minimum set of machinery is required for various ideas. So, for example, various properties of categories can be understood in the context of partially ordered sets (posets), so these ideas are first introduced for posets.
He is considering mathematics as a branch of computer science. Again using the minimum tools required, where possible he gives "intuitionistic" or "constructive" proofs of the existence of mathematical objects. Such proofs can be translated into programs that produce an instance with the properties required, linking proof and computability.
He is also considering mathematics as a human activity. It's not enough to give a foundation of mathematics, he also wants to describe how mathematicians actually proceed and show that the somewhat informal language of mathematics as actually practised can be translated into something rigorous.