I'm stuck with the following problem: Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be scheme morphisms such that f is surjective and universally closed and such that $g \circ f$ is separated. The claim is then that g is also separated.
I've been trying to use the fact that f is (as any surjective morphism) universally surjective, and somehow deduce that the diagonal $\Delta(Y)$ is closed in $Y \times_Z Y$ but I haven't gotten that far. I would love some hints on how to do this. Full answers are OK, but I would prefer to have hints! Thank you!