You let the cat outside, and changed its litter-box.
You seem to have a really wrong mental model of how fecal-oral transmission happens.
If I have Norwalk (Norovirus) someone else doesn't need to literally lick my butt or eat my feces to get it. If I wandered around their apartment naked (interesting image) and sat on or walked across random pieces of furniture, they'd be very likely to get it.