TITLE: Abstract nonsense proof
QUESTION [10 upvotes]: What is a simple example of an "abstract nonsense" proof in category theory. For a theorem you are proving, it doesn't matter if the category or regular proof came first, it is just that the category one should be simpler or more elegant. Note that it doesn't matter if the non-category prerequisites are some what complicated. I am sort of looking of how category theory can transfer proofs from one field to another. It should be at a rather simple level if possible.

REPLY [7 votes]: Well, there's Lawvere's rather nice proof of a form of Cantor's theorem; namely, if an object $X$ can index all its own morphisms $X\to A$, then every endomorphism of $A$ has a fixed point.
Assume $e:X\times X\to A$ has the universal property of an evaluation morphism, and let $f$ be any morphism $A\to A$. We form the composite $m:=f\circ e\circ\langle id_X, id_X\rangle:X\to A$. By the universal property for $X$ and $e$, there's a unique morphism $q:1\to X$ with $e\circ \langle id_X, q!\rangle=m$. If we do a little arrow chasing, however, we can see that $m\circ q=f\circ m\circ q$, and $f$ therefore has a fixed point; but $f$ was arbitrary, so any $f$ would have a fixed point.