Now that we have a coevaluation for vector spaces, let’s make sure that it intertwines the actions of a Hopf algebra. Then we can finish showing that the category of representations of a Hopf algebra has duals.
Take a representation , and pick a basis of and the dual basis of . We define the map by . Now , so if we use the action of on before transferring to , we get . Be careful not to confuse the counit with the basis elements .
On the other hand, if we transfer first, we must calculate
Now let’s use the fact that we’ve got this basis sitting around to expand out both and as matrices. We’ll just take on matrix indices on the right for our notation. Then we continue the calculation above:
And so the coevaluation map does indeed intertwine the two actions of . Together with the evaluation map, it provides the duality on the category of representations of a Hopf algebra that we were looking for.