Properness cancels through a separated morphism
= Properness cancels through a separated morphism
If $X\xrightarrow fY\xrightarrow gZ$ has proper composite and $g$ is separated, the graph $X\to X\times_ZY$ is a closed immersion, being a base change of the diagonal of $g$. The projection $X\times_ZY\to Y$ is a base change of the proper composite. Their composition is $f$, so $f$ is proper.