Product of Chu spaces in the category of $Chu(S-Act)$

The necessary conditions for the existence of products of Chu spaces in the category $Chu(S-Act)$ over the category of S-acts are obtained. It is proved the existence of a product of Chu spaces $r_i: A_i\otimes X_i\longrightarrow D_i$ $(i\in I)$ in the category $Chu (S-Act)$ in the cases of $X_i$ are zero polygons for any $i\in I$ or $D_i$ are zero polygons for any $i\in I$.