How does catdog?

Answer 1

Maybe they poop out of each other's mouth

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There's a theory circulating around in many underground groups called the "Third Hole Theory" it suggests that between Cat and Dog there is a third hole which all the poop comes out. This still only remains a theory as it was never seen in the show.

Alternatively - Cat coughs it up as a hairball - analogous to owl pellets. If true, this still would not address the question of how Catdog urinates.