Given three geometrically irreducible normal $ k$ -curves, $ X$ , $ Y$ , $ Z$ , and two morphisms $ f \colon\ X \to Z$ , $ g \colon\ Y \to Z$ . Assume that $ X \times_Z Y$ is irreducible. Does $ X \times_Z Y$ is geometrically irreducible? If not, which conditions should $ f$ and $ g$ satisfy? I am interested in case that $ f$ and $ g$ are étale morphisms.

More general form of the above question: Assume $ X$ , $ Y$ , $ Z$ , $ f$ and $ g$ as above. Let $ W$ be an irreducible component of $ X \times_Z Y$ . Does $ W$ is geometrically irreducible?

Thank you!