Natural isomorphisms of the forgetful functor

Question

Let $U: \mathbf{Groups} \rightarrow \mathbf{Sets}$ be the forgetful functor. Must every natural transformation $\eta: U \rightarrow U$ be a natural isomorphism?

Answer 1

Since $U$ is represented by $(\mathbb{Z},+)$, we have (by Yoneda) $\hom(U,U) \cong U(\mathbb{Z},+)=\mathbb{Z}$. If $z$ is an integer, the corresponding natural transformation $U \to U$ is given by $U(G) \to U(G),~g \mapsto g^z$ for groups $G$. From this it can be checked that actually we have an isomorphism of monoids $(\hom(U,U),\circ) \cong (\mathbb{Z},*)$. Since the only invertible elements of $(\mathbb{Z},*)$ are $\pm 1$, there are lots of natural transformations which are no isomorphisms, for example $g \mapsto g^2$.