A proper subcontinuum $H$ of a continuum $X$ is said to be a $W$-set provided for each continuous surjective function $f$ from a continuum $Y$ onto $X$, there exists a subcontinuum $C$ of $Y$ that maps entirely onto $H$. Hereditarily weakly confluent (HWC) mappings are those with the property that each restriction to a subcontinuum of the domain is weakly confluent. In this paper, we show that the monotone image of a $W$-set is a $W$-set and that there exists a continuum which is not in class $W$ but which is the HWC image of a class $W$ continuum.