Composing that adjoint from the one giving the free algebra on a vector space you'd end up with a free Hopf algebra on a vector space. Those don't exist, cf. http://ncatlab.org/nlab/show/free+Hopf+algebra
Later: Ah! Anton wants an adjoint on the other side, so this is not an answer. I'll just leave it here, because anyways it is a cute piece of information.