Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \to H$ which is also a left $H$-comodule map. It seems to be that such things cannot exist, but I can see a good way to prove it . . . For example if we assume that the subcoalgebras of $H$ are non-isomorphic and that $H$ is generated by the elements of a single subcoalgebra, then the result seems to work . . . but it is not clear to me that these conditions are necessary.