For an algebraically closed field k $, let $ C $ be a k $-coalgèbre. Given a minimal injectable 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 have not found any elaborate examples.

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

The polynomial ring is a good starting point for me. $ k (x_ {ij}) $ with coproduct $$ Delta (x_ {ij}) = sum_k , x_ {ik} x_ {kj}, $$

but i am happy with all kinds of examples / references.