I think so. Let $f$ be a homomorphism from $\mathbb Z[[x]] $ to $\mathbb Z$. WRONG: <strike> Let $f(x^i)=a_i$. Then since $f(1+x+x^2...) \in \mathbb Z$, we must have $a_i=0$ for $i\gg 0$. So each map can be identified with an element in $\mathbb Z[x]$.</strike> 

An attempt at redemption: I actually found a reference on when the dual of direct product of a ring is direct sum: 

http://www-users.mat.umk.pl/~gregbob/seminars/2008.11.07b.pdf