Internal hom for modules over a Hopf algebra ($(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$)

The full ${\mathrm{Hom}}_k(M,N)$ carries the action $(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$, making it the internal hom right adjoint to $-\otimes M$. No finite-dimensionality or inverse antipode is needed.