Inducing a comodule structure on Hom If C is an R-coalgebra and M is an R-module... then is it possible to endow $Hom_{_RMod}(C,M)$ or $Hom_{_RMod}(M,C)$ with the strucutre of a C-comodule?

_Hint._ We have a canonical span $\hom_R(M,C) \to \hom_R(M,C \otimes_R C) \leftarrow \hom_R(M,C) \otimes_R C$. The right arrow is an isomorphism when $M$ is finitely generated projective (first check $M=R$, then direct sums, then direct summands).

I don't see such a construction for $\hom_R(C,M)$.