There seem to be two types of undergraduate exercises in set theory: the boring ones (e.g. where we are asked to compute the intersection of an indexed family of sets, or draw a diagram of a relational composition) and the exciting ones (where we ask why a set cannot be of the same size as its powerset, or why having two injections f:A->B and g:B->A implies that A and B are isomorphic). The difference is of course that the first kind involves mechanical computation with points, and all data given, and the second kind needs a creative argument in a situation where it sometimes seems that there is not enough data to solve the problem.
The book by Lawvere and Rosebrugh made me realise that the exercises I find exciting can be phrased in terms of properties of maps acting on sets, and - yes, indeed - the boring ones can't.
But then the book goes further - it shows that in fact all axioms of sets can be written down in the language of maps. Doesn't it strike you that as a consequence axiomatic set theory is not boring? :)
Some of the discoveries I made during reading are just invaluable. For example, I learned that the *reason* why a set cannot be of the same size as its powerset is that the two-element set have a self map with no fixed point, which is, admittely, the essence of Cantor's diagonal argument.
The Authors say in the Foreword that the book is for students who are beginning the study of algebra, geometry, analysis, combinatorics, ... Indeed, being a virtuoso of a particular implementation of set theory such as ZFC does not help much with these subjects. Instead one needs a good knowledge of how sets behave when measured, divided, added, towered, counted - name your favourite operation - and this is precisely the story told in the book.