  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.
