Dual of coideals

Question

Let $C$ be a coalgebra with comultiplication $\Delta$ and count $\epsilon$. A coideal $I$ in $C$ is a linear subspace such that $$\epsilon(I)=0 \qquad \text{and} \qquad \Delta(I)\subset I\otimes C +C\otimes I\,.$$ Notice that the linear dual $C^\ast$ of $C$ is naturally an algebra (no finite dimension required). My question is that what is the dual of $I$ in $C^\ast$? I don't think it would be an ideal in $C^\ast$. Thanks in advance.