ct.category theory - Simple show cases for the Yoneda lemma

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 $(V_G, E_G, S_G:Erightarrow V, T_G:E rightarrow V)$.

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?