In p. 18 of Categories for the working mathematician (2d ed.), Mac Lane remarks that
…"category" has been defined in order to define "functor" and "functor" has been defined in order to define "natural transformation".
For a concept to warrant so much preliminary machinery it must yield huge benefits, but somehow in the case of natural transformations I still don't see them.
For example, one of the few examples of natural transformations that I can fully follow among those that Mac Lane offers is the determinant natural transformation, $\det$ (on p.16). His description goes something like this.
Let $\mathbf{CRng}$ and $\mathbf{Grp}$ be, respectively, the categories of commutative rings and of groups; fix $n$ to be an arbitrary natural number; for any commutative ring $K$, let $\mathrm{GL}_nK$ be the group of all non-singular $n \times n$ matrices with entries in $K$, and let $K\text{*}$ be the group of invertible elements (aka units) of $K$. Then, $\mathrm{GL}_n$ and $\text{*}$ are functors $\mathbf{CRng}\to\mathbf{Grp}$, and, moreover, the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
\mathrm{GL}_nK @>\det_K>> K\text{*}\\
@V\mathrm{GL}_nfVV @VVf\text{*}V\\
\mathrm{GL}_nK^\prime @>\det_{K^\prime}>> K^\prime\text{*}
\end{CD}
$$
IOW, $\det$ is a natural transformation $\mathrm{GL}_n \dot{\to} \text{*}$.
It's a very nice, very clear example, but after I follow all the definitions and chase all the arrows, I'm left asking myself "so what"?
More specifically, before working out the diagram, I knew pretty much all the information expressed by its arrows, and compositions thereof. Even the commutativity between "taking the determinant" and "applying $f$" is not hard to see. So, at least in this case, the concept of natural transformation does not seem to be contributing much to what I already knew, and it's thus hard for me to justify all the apparatus needed to define it.
What am I missing?
Best Answer
The naturality squares of a natural transformation are: $$\require{AMScd} \begin{CD} FA @>{\eta_A}>> GA\\ @V{Ff}VV @VV{Gf}V\\ FB @>>{\eta_B}> GB \end{CD}$$ where we ask the square to commute for any choice of $f\in \mathsf{Hom}(A,B)$.
To me, the significance of this definition is somewhat philosophical (and has hitherto given me good intuition for naturality):
By the Yoneda lemma, $A\cong B\iff H_A\cong H_B$ where $H_A=\mathsf{Hom}(-,A)$. This means that objects are determined by the arrows into them. Asking the naturality squares to commute for every appropriate $f$ is basically saying "this square commutes regardless of what $\mathsf{Hom}(A,B)$ looks like", which by Yoneda is the same as saying "this square commutes regardless of the properties of $A,B$".
Thus, to say some arrow $\eta _A$ is natural in $A$ is to say that the specific object $A$ plays no role in defining it, hence it can be extended to a global construction between functors.
The most elegant example I can think of where naturality really simplifies a proof is the homotopy invariance of singular homology. In this instance, naturality literally enables you to solve a global problem (for all topological spaces) by solving it only for very simple spaces (the standard simplices). This same principle is the heart of the acyclic models theorem, which itself can be used to prove many other wonderful theorems.