TITLE: Simple show cases for the Yoneda lemma
QUESTION [18 upvotes]: I've been given a very simple motivating and instructive show case for the Yoneda lemma:
Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ T_G:E \rightarrow V)$. 
Consider $K_1$ and $K_2$, the one-vertex and the one-edge graph and the two morphisms $\sigma$ and $\tau$ from $K_1$ to $K_2$.
Now consider the graph $H$ with

$V_H = Hom(K_1,G)$
$E_H = Hom(K_2,G)$
$S_H(e) = e \circ \sigma: K_1 \rightarrow G$ for $e \in E_H$
$T_H(e) = e \circ \tau: K_1 \rightarrow G$ for $e \in E_H$

It can be easily seen that $H$ is isomorphic to $G$.
I have learned that a) the category of graphs is a presheaf category and that b) $K_1$, $K_2$ are precisely the representable functors.
Now I am looking for other simple motivating and instructive show cases.
By the way: Shouldn't such an show case be added to the Wikipedia entry on Yoneda's lemma?

REPLY [3 votes]: Not so much a showcase maybe but rather a connection to an algebraist's intuition. A particular case of Yoneda is that a monoid $M$ acting on itself (by (say) left multiplication) is a free $M$-set on one generator, $\hom_M(M,X)\approx X$ naturally in $X$.
This can be in fact of course extended to the full Yoneda lemma - viewing set-valued functors on a category $\mathcal C$ as algebras for a multi-sorted equational theory (objects of $\mathcal C$ for sorts, unary operations only), Yoneda lemma becomes the statement that representables are single-generator free algebras.