Functional analytic issues in ${\mathbb{Z}}_2^n$-Geometry

We show that the function sheaf of a ${\mathbb{Z}}_2^n$-manifold is a nuclear Fr'echet sheaf of ${\mathbb{Z}}_2^n$-graded ${\mathbb{Z}}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a ${\mathbb{Z}}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of ${\mathbb{Z}}_2^n$-manifolds. All proofs are self-contained and explicit.