Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th symmetric powers. Is $f^{[n]}$ also etale? Clearly, affine $X$ and $Y$ suffice for the purposes of the question.

It seems from VA.'s answer to Smoothness of Symmetric Powers that the answer is yes, but his argument is too terse for me; in particular, I don't understand how to harmlessly pass to formal completions. I would be grateful if someone could give a more detailed explanation.

EDIT. As abx noted in the comments, my question is based on a false expectation. Give this, let me modify the question: how does one prove that $X^{[n]}$ is $k$-smooth by reducing to the $X = \mathbb{A}^1_k$ case? That is, how to carry out the reduction?