For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called *basic coalgebra* $B_C=coend^C(E)$, s.t. the comodule categories $Mod^C$ and $Mod^{B_C}$ are equivalent. I would like to understand this object $B_C$ better, but I didn't find any elaborated examples.

I am particularly interested in the case when $C$ is a coquasitriangular Hopf algebra. In this case, is $B_C$ still a (coquasitriangular) Hopf algebra, so that we have a (braided) monoidal equivalence of categories?

A good starting point for me would be the polynomial ring $k[x_{ij}]$ with coproduct $$\Delta(x_{ij})=\sum_{k=0}^n\,x_{ik}\otimes x_{kj},$$ but I am happy with all kinds of examples/references.