I agree completely with J. Elliott. The author states so many propositions without proof, and even the proofs given are too sketchy, forcing the reader to fill in every detail, and in many instances, the author's proofs are simply wrong. Many of his definitions are vague and confusing, in many cases bewildering the reader's mind with all kinds of tangential questions unrelated to the main topic. Paul Taylor misleads the reader with chapter titles like "Posets and Lattices" and "Cartesian Closed Categories" in which he does not stick to the topics he promises to cover but jumps all over the place into unrelated fields. It's like he wants to "introduce" the reader to so much that he has no time to explain anything.
Besides, there are so many better books for any of the subjects the book brings up. For category theory, there is "Categories for the Working Mathematician" by MacLane; for lambda calculus, there is Barendregt's, for topos theory, there is "Topoi" by Goldblatt, who does not prove everything he states, including several fundamental theorems, but at least he stays on topic; or if one simply wishes to forget about new approaches to foundations and take up traditional set theory, there is Jech, whose book is very difficult, but at least it it challenging. But as for Taylor, his is neither interesting, nor enlightening, nor even challenging. As for those who already "know it all", what's the point?
In short, the author does not start with the basics and build up in any sort of cumulative fashion, but diverts the reader's interests into every specialization into which mathematics is expanding. "A practical foundation of mathematics" is anything but foundational. Tempus est legendi aliud.