Most general form of Cayley's theorem?
Just as Cayley's theorem tells us that any group can be embedded in the symmetric group $\operatorname{Sym}G$, the Yoneda lemma of category theory says that any category $D$ can be embedded into a category of functors.
It turns out Cayley's theorem can be viewed as a special case of the Yoneda lemma. There is a way to view a group as a category, and in this context the Yoneda embedding can be seen to be the Cayley embedding.