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.