Are there any (reasonable) conditions (imposed on the maps) that guarantee that the converse is also true? The motivation behind this question is the following (for example), coming from Algebraic Geometry: "

*Let $f \ \colon X \longrightarrow Y$ and $g \ \colon Y \longrightarrow Z $ be morphisms of (noetherian) schemes. If $ g \circ f $ is proper and $g$ is separated, then $f$ is also proper*."