It presupposes a certain degree of mathematical maturity, but Robert Geroch's _Mathematical Physics_ (don't let the title fool you!) has the most intuitive explanations (with diagrams) I've ever seen of definitions, concepts, and proofs. It's roughly at the first-year grad level, but is almost completely self-contained, and uses category theory to motivate the entire organization of the book (whatever object is being treated in the current chapter generally has a forgetful functor to an object in the previous chapter).