On this page
https://ncatlab.org/nlab/show/rigid+monoidal+category
there is a discussion of rigidity (left-right duality) for the catagory of modules over a Hopf algebra. What happens if we look at the category of comodules over a Hopf algebra? Do we still get a rigid category? If so, does there exist any general relationship between the left and right duals?
The category of the finite dimensional comodules of a hopf algebra over a field, is a rigid, monoidal category. (just as the category of the finite dimensional modules).
If we take a fin dim, right comodule $X$, of the hopf algebra $H$ then its dual vector space $X^*$, equipped with the unique comodule structure satisfying $e_V(\phi_{(0)}\otimes v)\phi_{(1)}=e_V(\phi\otimes v_{(0)})S(v_{(1)})$ provides the corresponding left dual object, i.e. the right $H$-comodule $X^*$ which is the left dual of the right $H$-comodule $X$. (I am not sure however for the last part of your question: if there exists some general relationship between left and right dual objects).
In fact, more can be shown. In On hopf algebras and rigid monoidal categories, Isr. J. of Math. v.72, p.252–256, (1990), Ulbrich has shown that:
An ordinary bialgebra $H$, over a field $k$, is a hopf algebra if and only if its category of finite dimensional comodules is rigid.
There, the author essentially uses the dual object to "reconstruct" an antipode for the original bialgebra.
