  Let $X$ be an affine scheme of $k \times \mathbb{N}$-matrices and $Y$ be an
affine scheme of $\mathbb{N} \times \cdots \times \mathbb{N}$-dimensional
tensors. The group Sym$(\mathbb{N})$ acts naturally on both $X$ and $Y$ and on
their coordinate rings. We show that the Zariski closure of the image of a
Sym$(\mathbb{N})$-equivariant morphism of schemes from $X$ to $Y$ is defined by
finitely many Sym$(\mathbb{N})$-orbits in the coordinate ring of $Y$. Moreover,
we prove that the closure of the image of this map is
Sym$(\mathbb{N})$-Noetherian, that is, every descending chain of
Sym$(\mathbb{N})$-stable closed subsets stabilizes.
