A creation like a slideshow is dependent on a list of pictures in your catalog as well as a list of assets on your computer. Those items are identified by links to the actual location (path) of your real pictures. Even if you copy the whole catalog in another computer, the links won't work on a different drive/computer.
The only way I can see to work on the same slideshow project alternatively on two computers is the following:
- you move your picture file folder tree to an external drive
- you move your catalog to the same drive (in 'custom' location)
- you plug in the external drive to both computers alternatively. Your catalog will be recognized by both computers and it will stay updated.