I’m not sure why, but this comment by Easwaran reminded me that the book Toposes, Triples, and Theories, by Michael Barr and Charles Wells, is available for downloading, if your vices run in that particular direction. A topos is a category-theoretic analogue of a set theory. The category of sets for a topos, but there many others. A triple (now usually called a monad) is a category-theoretic analogue of an algebra (in the sense of universal algebra). I don’t remember what a theory is.