Yoneda Lemma in 2-category. I have seen Yoneda's Lemma and some simple applications such as the functor of points and also that it is a generalization of Cayley  theorem for groups. My question: Can Cayley  theorem be stated in another way if we work with Yoneda's lemma in a 2-category?
Thank you very much
 A: Cayley's theorem states that every group may be embedded in the automorphism group of a set. Turning everything a dimension up (and keeping it strict for simplicity) we do expect that every 2 group may be embedded in the automorphism 2 group of a category.
Given a 1 category $C$, we get a 2 category Aut($C$) of invertible functors $f:C\to C$ and invertible 2 cells between them. A 1 group is a 1 object category in which every morphism is invertible. So let us define a (strict) 2 group to be a one object (strict) 2 category in which all 1 and 2 cells are invertible. Then Aut($C$) of a 1-category is such a strict 2 group.
Now let $G$ be an arbitrary 2 group. The strict 2 Yoneda lemma gives us an embedding $y:G\to 2Fun(G^{op},Cat)$ of $G$ into the 2 category of 2 functors, 2 natural transformations and modifications. Finally, we may postcompose with evaluation at the single object $*$ of $G$ to get a 2 functor $G \to 2Fun(G^{op},Cat) \to Cat$ whose images sits inside the 2 group Aut($G(*,*)$). The two functor sends a 1 morphism $g$ of $G$ to the functor $G(*,g)$ and a 2 cell $\beta: g \to g'$ to the natural transformation $G(*,\beta): G(*,g) \to G(*,g')$.
Since $G$ has only one object evaluating at * is an embedding, and hence we have embedded $G$ into a 2 group of automorphisms of a category. A good name for the result is the strict 2 Cayley theorem.
