Quark Catalyst is a Macintosh Finder-like GUI desktop for the Apple IIe. I have not been able to try this software myself, but a while back these screen photos surfaced. Hats off to Blake Patterson who managed to acquire this rare piece of software and posted these photos of it in operation on his Flickr page.

Catalyst 3 was produced around 1985 by Quark, inc which is more famous for another product: Quark Express. It was bundled with early Apple 3.5" UniDisk floppy drives for the Apple IIe. The original floppy disks contain a form of copy "protection" that apparently prevents it from being read in later 3.5" drives, and no version is known to exist that can be used in emulators.

Catalyst was primarily a tool for running 5.25" floppy disk based applications from a hard disk drive or the higher capacity 3.5" disks. (Many 5.25" disk Apple software of the day used custom self-contained disk OSes that often did not inter operate with the standard AppleSoft DOS or ProDOS.) Versions 1 and 2 were text-mode based, version 3 added the Mac like desktop.

At the time it competed with MouseDesk (Apple II Desktop) and looks very similar, but the two are not related.

This is the "About" screen. Like the Macintosh desktop, icons representing the disk drives appear directly on the desktop as does a "Trash" icon. Although not specifically an Apple product, it uses an Apple logo for the first menu item.
 

Files are displayed as icons in folder windows. Different icons are displayed based on the ProDOS file types.

It is hard to tell from these screen shots if file management, such as dragging and dropping, works like one would expect.

A few of the menu items. It is not clear if its ability to load disk-based software to hard disks is built directly in to the desktop or separate functionality.
 

Of course it mimics several of the Macintosh accessories. Shown here are a calculator, clock, and sliding puzzle.

And a couple additional photos, first a brief advertisement for Catalyst and another one of Quark's products "Word Juggler".

A photo of what appears to be an earlier version of Quark Catalyst.

Finally, a Quark Catalyst 3.5" floppy disk.