Given a graded coalgebra $C = \bigoplus_{n\geq 0} C_n $ with coproduct
$$\Delta : C_n \to \bigoplus_{i=0}^n C_i\otimes C_{n-i} $$
must we have that the dual $C^* = \bigotimes_{n\geq 0}C_n^*$ is a graded algebra i.e. the product is a map
$$ C_i^*\otimes C_j^* \to C_{i+j}^* $$