Ventilating a cat box through the dryer exhaust duct
Sharing ducts in that way usually isn't allowed by code for health, comfort, energy efficiency, and fire reasons, among others. It's not a good idea, as you won't balance venting force without a diverter of some sort, and that means you have to constantly be switching it back and forth for the current need. You're almost guaranteed to forget at times, which creates fairly obvious problems.
Run a second duct along the first. Install a new vent outside with an operational backflow/draft preventer flap.