(Alternate title: *Find the Adjoint: Hopf Algebra edition*)


I was chatting with Jonah about his question [Hopf algebra structure on $\prod\_n A^{\otimes n}$ for an algebra $A$](https://mathoverflow.net/questions/22536/hopf-algebra-structure-on-prod-n-a-otimes-n-for-an-algebra-a). It's very closely related to the following question:



> 
> For a $k$-algebra $A$, is there a Hopf Algebra $H(A)$ such that for any Hopf algebra $B$, we have  
> 
>  $\mathop{Hom}\_{k\text{-alg}}(B,A)\cong \mathop{Hom}\_{k\text{-Hopf-alg}}(B,H(A))$?
> 
> 
> 


In other words, does the forgetful functor from $k$-Hopf-algebras to $k$-algebras have a right adjoint? There are a few related questions, any of which I'd be interested in knowing the answer to:


* Does the forgetful functor from Hopf algebras to *augmented* algebras (sending the counit to the augmentation) have a right adjoint?
* Does the forgetful functor from Hopf algebras to algebras with distinguished anti-automorphism (sending the antipode to the anti-automorphism) have a right adjoint?
* Does the forgetful functor from Hopf algebras to algebras with augmentation *and* distinguished anti-automorphism have a right adjoint?


Unfortunately, I don't feel like I can motivate this question very well. My motivation is that the better I know which forgetful functors have adjoints, the better I sleep at night.



Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint?
The article [here](http://arxiv4.library.cornell.edu/pdf/0905.2613) proves that the forgetful functor from $k$-Hopf algebras to $k$-algebras has a right adjoint. The main tool they use is the Special Adjoint Functor Theorem. 


