The Stacks project

Example 68.18.1. If $X, Y, Z$ are schemes, then the set $F_{x, z}$ is equal to the spectrum of $\kappa (x) \otimes _{\kappa (y)} \kappa (z)$ (Schemes, Lemma 26.17.5). Thus we obtain a finite set if either $\kappa (y) \subset \kappa (x)$ is finite or if $\kappa (y) \subset \kappa (z)$ is finite. In particular, this is always the case if $g$ is quasi-finite at $z$ (Morphisms, Lemma 29.20.5).