Properness cancels through a separated morphism

If $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ has proper composite and $g$ is separated, the graph $X\to X{\times }_{Z}Y$ is a closed immersion, being a base change of the diagonal of $g$. The projection $X{\times }_{Z}Y\to Y$ is a base change of the proper composite. Their composition is $f$, so $f$ is proper.