Let $G$ be a finite group of order $n$ and consider the category $\mathscr G$ with a single object $\{ \cdot \}$ and whose morphisms consist of the elements of $G$, i.e., $\mathrm{Hom}_{\mathscr G}(\cdot, \cdot)=G$. 

Let $h^{\cdot}=\mathrm{Hom}_{\mathscr G}(\cdot , {\_} )$ be the Yoneda functor. Then according to Yoneda's lemma
$$\mathrm{Nat}(h^{\cdot},h^\cdot)\simeq \mathrm{Hom}_{\mathscr G}(\cdot,\cdot),$$
which implies that $h^\cdot$ is a faithful functor. In other words, 
$$ h^{\cdot} : \mathscr G \to \mathscr Set$$
maps $\cdot$ to the set $\mathrm{Hom}_{\mathscr G}(\cdot,\cdot)\simeq_{\mathscr Set} |G|$ (where $|G|$ is the set of elements of $G$)
and embeds the group (!) $\mathrm{Hom}_{\mathscr G}(\cdot,\cdot)\simeq_{\mathscr Groups} G$ into $\mathrm{Hom}_{\mathscr Set}(|G|,|G|)\simeq S_n$, which gives you Cayley's Theorem.