Is the fiber product $X\times_Y Z$ locally irreducible when $X$, $Y$, and $Z$ are all locally irreducible?

Question

The space $X$ is called irreducible at $x$ if the stalk $\mathcal{O}_x$ of its structure sheaf is an integral domain.

The space $X$ is called locally irreducible if all points of $X$ are irreducible.

Assume that $X$, $Y$, and $Z$ are all locally irreducible complex analytic spaces. Furthermore, $f:X\to Y$ and $g:Z\to Y$ are two holomorphic maps (morphisms). Then is the fiber product $X\times_Y Z$ locally irreducible?

Answer 1

No. Take, for instance $$ X = Y = Z = \mathbb{A}^1 $$ with the maps $X \to Y$ and $Z \to Y$ given by $x \mapsto x^2$ and $z \mapsto z^2$, respectively.