This "Hat Conformator & Stretcher" was invented by Hugh Lyons of Lansing, Michigan. It was produced by the Hugh Lyons & Company, also of Lansing, ca.1890. As its name suggests, the device did two things. It stretched the hat to a particular size and it also conformed the hat to the specific dimensions of the wearer's head.
The Lyons was just one of several now-antique hat conformators (sometimes spelled conformateurs), hat stretchers and hat sizers that were produced. I'm interested them all. The more peculiar and complicated the apparatus, the more I'm interested.