Products in the category of ????2n-manifolds

We prove that the category of $\mathbb{Z}_2 ^n$-manifolds has all finite products. Further, we show a ^n$-manifold (resp., ^n$-morphism) can be reconstructed from its algebra global ^n$-functions morphism between ^n$-function algebras). These results are importance in study ^n$ Lie groups. The investigation is more challenging, since completed tensor product structure sheafs two not sheaf. rely on number (pre)sheaves topological algebras, which establish appendix.