TITLE: Cayley's Theorem and the Yoneda Lemma
QUESTION [10 upvotes]: Hi, from wiki, I know that yoneda lemma is the generalization of Cayley's theorem. But I am not quite understand the intuition behind that. Anyone can help me with that? Cheers!

REPLY [10 votes]: Let $G$ be a finite group of order $n$ and consider the category $\mathscr G$ with a single object $\{ \bullet \}$ and whose morphisms consist of the elements of $G$, i.e., $\mathrm{Hom}_{\mathscr G}(\bullet, \bullet)=G$. 
Let $h^{\bullet}=\mathrm{Hom}_{\mathscr G}(\bullet , {\_} )$ be the Yoneda functor. Then according to Yoneda's lemma
$$\mathrm{Nat}(h^{\bullet},h^\bullet)\simeq \mathrm{Hom}_{\mathscr G}(\bullet,\bullet),$$
which implies that $h^\bullet$ is a faithful functor. In other words, 
$$ h^{\bullet} : \mathscr G \to \mathscr Set$$
maps $\bullet$ to the set $\mathrm{Hom}_{\mathscr G}(\bullet,\bullet)\simeq_{\mathscr Set} |G|$ (where $|G|$ is the set of elements of $G$)
and also embeds the group (!) $\mathrm{Hom}_{\mathscr G}(\bullet,\bullet)\simeq_{\mathscr Groups} G$ into $\mathrm{Hom}_{\mathscr Set}(|G|,|G|)\simeq S_n$, which gives you Cayley's Theorem.