I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a higher, unified level. Categorical methods are general, but of course they can be applied to specific categories and thereby help us to solve specific problems. I am not asking for canonical applications in which category theory is used. I have read all answers to similar math.SE questions on applications of category theory, but they don't fit to my question below. I would like to ask for applications of the notions of "category", "functor", and "natural transformation" (perhaps also "limit" and "adjunction") , which go beyond descriptions, but really solve specific problems in an elegant way. I am aware of many, many proofs of theorems which have category-theoretic enhancements, in particular by means of the Yoneda Lemma, but I'm not looking for these kind of applications either. So my question is (even though I know that this is not the task of category theory):
Can you name a specific and rather easy to understand theorem, whose statement naturally does not contain any categorical notions, but whose proof introduces a suitable category / functor / natural transformation in a crucial way and uses some basic category theory? The proof should not just depend on a large theory (such as arithmetic geometry) whose development has used category theory over decades. The proof should not just be a categorical version of a proof which was already known.
So here is an example of this kind, taken from Hartig's wonderful paper "The Riesz Representation Theorem Revisited", and hopefully there are more of them: Let $X$ be a compact Hausdorff space, $M(X)$ the Banach space of Borel measures on $X$ and $C(X)^*$ the dual of the Banach space of continuous functions on $X$. Integration provides a linear isometry
$$\alpha(X) : M(X) \to C(X)^*, ~ \mu \mapsto \bigl(f \mapsto \int f \, d\mu\bigr).$$
The Riesz Representation Theorem asserts that this is an isomorphism. For the "categorical" proof, observe first that the maps $\alpha(X)$ are actually natural, i.e. provide a natural transformation $\alpha : M \to C^*$. Using naturality and facts from functional analysis such as the Hahn-Banach Theorem, one shows that if $X$ satisfies the claim and admits a surjective map to $Y$, then $Y$ satisfies the claim. Since every compact Hausdorff space is the quotient of an extremally disconnected space, namely the Stone-Cech-compactification of its underlying set, we may therefore assume that $X$ is extremally disconnected. Now here comes the actual mathematics, and I will just say that there are enough clopen subsets which allow you to construct enough continuous functions. The general case has been reduced to a very easy one, using the concept of natural transformation.
Best Answer
Does the following standard proof of the Brouwer fixed point theorem for the two-dimensional disk $D$ count?
Theorem. Any continuous map $f : D \to D$ has a fixed point.
Proof. If $f$ had no fixed point, the map $g : D \to \partial D$ given by $g(x) = \partial D \cap ($ray from $f(x)$ to $x)$ would be a retraction of $D$ onto $\partial D$, that is, $g \circ i = 1_{\partial D}$ where $i : \partial D \to D$ is the inclusion. This implies, by functoriality of $\pi_1$, that $g_\ast \circ i_\ast = 1_{\pi_1(\partial D)}$ which is impossible since $\pi_1(D) = 0$, $\pi_1(\partial D) = \mathbb{Z}$.