i'm looking for a beginner's example to a somewhat non-obvious insight one can get by formalising things using category theory. such things clearly exist among many areas of pure math but i am looking for examples using the "simpler" categories Set or Vectk (or the relationship between them) because those are mathematical objects that are easier for me to understand than rings, groups and other advanced objects that are often used as examples for professional mathematicians.
to give some motivation: as a beginner i can see the value of categories and diagrams but it looks like a lot of the work in coming up with the diagrams is done "outside" category theory. for instance if i have:
$X \rightarrow^{f} Y$
$Y \rightarrow^{g} Z$
$X \rightarrow^{h} Z$
where $X, Y, Z$ are sets and $f, g, h$ are functions, then it's helpful to note if the diagram commutes, and $(g \circ f)(X) = h(X)$. but to know that for any specific set of $f, g, h$ that operate on sets one has to prove that by reasoning about sets - then if the diagram commutes this can be concisely communicated. i see the value in this.
however, is there an example of how one can write down simple relationships about sets in the form of diagram and deduce something somewhat non-obvious by reasoning about the diagrams? similarly, is there an example of a non-obvious relationship between operations in Vect and in Set?
For an example of the power of the language of diagrams:
Namely, in the same sense as $\min$ is dual to $\max$, or $\gcd$ is dual to $\mathrm{lcm}$: we just have to formally revert the arrows in the definition.
Consider the category $Set^{fin}$ of finite sets and functions between them (so it's a full subcategory of $Set$), and two partial orders $\ \le\, $ and $\,|\ $ on $\Bbb N$, considered as categories (there's a unique arrow $n\to m$ iff $n\le m$ [resp. $n|m$]).
We say that an object $P$ (together with arrows $\pi_1,\pi_2$) is a product of objects $A,B$ in a category, if $\pi_1:P\to A,\ \pi_2:P\to B$, and any similar diagram targeted at $A, B$ uniquely factors through $\pi_1,\pi_2$, i.e. for any $\tau_1:T\to A,\ \tau_2:T\to B$ there is a unique $\varphi:T\to P$ such that $\tau_1=\pi_1\circ\varphi$ and $\tau_2=\pi_2\circ\varphi$.
Note that, by the uniqueness criterion, if $P'$ is also a product of $A$ and $B$, then $P'\cong P$.
In the case of a poset category, this definition coincides with the 'greatest common precedor', so in the two specific categories above on $\Bbb N$, the categorical product of two numbers $n,m$ is just their minimum (for $\le$), and their $\gcd$ (for $|\,$).
In $Set$, as well as in $Set^{fin}$, a set $P$ is a product of $A$ and $B$ iff $P$ is isomorphic to the Cartesian product $A\times B$ (whence $\pi_1,\pi_2$ will be the projections [composed with the isomorphism]), or, equivalently iff $$|P|\ =\ |A|\cdot|B|$$ because sets are isomorphic iff they have the same cardinality.
To get the dual notion ('coproduct' or 'sum'), we simply revert the arrows in the definition of 'product'.
In a poset category, it clearly gives the dual notions: $\max$ for $(\Bbb N,\le)$ and $\mathrm{lcm}$ for $(\Bbb N,\,|) $.
In $Set$, $Q$ is a sum of $A$ and $B$ iff $Q$ is isomorphic to the disjoint union of $A$ and $B$ (considered together with the two inclusion maps) iff $$|Q|\ =\ |A|+|B|$$
Note also that in $Vect$, we have $A\times B\cong A+B$, so that (at least for finitely many objects) the sum and the product coincides here.