This question is, in some sense, a variant of [this][1], but for certain cases.

The opposite category of an abelian category is abelian. In particular, if $R-mod$ is the category of $R$-modules over a ring $R$ (say left modules), its opposite category is abelian.  The Freyd-Mitchell embedding theorem states that this opposite category can be embedded in a category of modules over a ring $S$. This embedding is usually very noncanonical though.

> **Question:** Is there any way to choose $S$ based on $R$?

My guess is probably not, since [these notes][2] cite the opposite category of $R-mod$
as an example of an abelian category which is not  a category of modules. I can't exactly tell if they mean "it   is (for most $R$) (provably) not equivalent to a category of modules over any ring" or "there is no immediate structure as a module category." If it is the former, how would one prove it?

I also have a variant of this question when there is additional structure on the category.

The module category $H-mod$ of a Hopf algebra $H$ is a tensor category.  A finite-dimensional Hopf algebra can be reconstructed   from the tensor category of finite-dimensional modules with a fiber functor  via Tannakian reconstruction.   Now $(H-mod)^{opp}$ is a tensor category as well satisfying these conditions (namely, the hom-spaces in this category are finite-dimensional).  The dual of the initial fiber functor makes sense and becomes a fiber functor from $(H-mod)^{opp} \to \mathrm{Vect}$ (since duality is a contravariant tensor functor on the category of vector spaces).  In this case, $(H-mod)^{opp}$ is the representation category of a canonical Hopf algebra $H'$.

> **Question$\prime$** What is $H'$ in terms of $H$?


  [1]: https://mathoverflow.net/questions/23361/construction-of-opposite-category-as-a-structure
  [2]: http://www.math.ku.dk/~alexb/HomAlg/Category.pdf