Products in the category of $\mathbb{Z}_2 ^n$-manifolds

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