TITLE: Which insights are particularly simple to get through category theory?
QUESTION [26 upvotes]: I'm trying to motivate myself to learn category theory, but I don't have a clear view of what would be the benefit. I've heard that it can be used to find similitudes between mathematical theories, but I haven't seen an example of this or maybe I can't even recognize them among the literature because I haven't mastered Category Theory to begin with. 
Therefore, I'm looking for an insight that requires very little knowledge on the subject (for motivation; you can't ask people to study a large subject without offering small rewards among the path) and that highlights its virtudes.

REPLY [4 votes]: Here's an interesting example.
You may be familiar with the notion of a graded ring. A common definition is that it is a ring whose additive abelian group can be written as a direct sum $A = \bigoplus_n A_n$ of abelian groups, where $n$ ranges over the integers, with the property that the product of something from $A_m$ with something from $A_n$ is an element of $A_{m+n}$.
I've read that the definition of a graded module over a graded ring had several candidate definitions, and it wasn't completely clear which one was right.

Now, enter the category-theoretic definition of module.
If $G$ is a group, we can construct the category $\mathcal{C}$ with one object and whose endomorphism monoid is precisely that group. A left $G$-set is a functor $\mathcal{C} \to \mathbf{Set}$. A left $G$-module is a functor $\mathcal{C} \to \mathbf{Ab}$.
Many of the good properties of the category of $G$-sets or the category of $G$-modules follow simply from general facts about functor categories, particularly $\mathbf{Set}$ or $\mathbf{Ab}$-valued functor categories.
Similarly, if $R$ is a ring, we can construct the preadditive category $\mathcal{C}$ with one object whose endomorphism ring is precisely that ring. A left $R$-module is a functor $\mathcal{C} \to \mathbf{Ab}$.
A graded ring $A$ has a description as a preadditive category $\mathcal{C}$ whose objects are natural numbers, with $\hom(m,n) = A_{n-m}$, and composition being the product of the ring. 
It is now obvious what the right definition of a left module over a graded ring really ought to be: a functor $\mathcal{C} \to \mathbf{Ab}$.