Let $X,Y,Z$ schemes with $f:X \to Y$ and $g:Z \to Y$ with $f$ faithfully flat. I'd like to prove that if $g_Y:X \times_{Y} Z \to X$ is a finite map , so it is $g$.
The part I'm having trouble with is showing that the map is affine: given that, proving that it is finite should be easier using faithfully flatness. I'm trying to use faithfully flat descent but I do not really do how to deal with this.