Given a Schwartz function $f(x,y)$ on $\mathbb{R}^{2n}$, is $f(x,x)$ necessarily a Schwartz functoin on $\mathbb{R}^n$?

Question

I am suddenly a bit confused by the question described in the title.

Let $f(x,y)$ be a Schwartz function on $\mathbb{R}^{2n}$ with $x,y \in \mathbb{R}^n$. Then, is it necessarily true that $x \to f(x,x)$ is a Schwartz function on $\mathbb{R}^{n}$?

In the case of $\mathcal{S}(\mathbb{R}^n) \otimes \mathcal{S}(\mathbb{R}^n)$ this seems to be the case.

However, I cannot see how to extend the result to whole $\mathcal{S}(\mathbb{R}^{2n})$. Could anyone please clarify for me?