Vaughan writes: How about the positive integers with * as sum, with 1->3 as the only nonidentity arrow? The unique cubical coalgebra is 1 but there is no square coalgebra. Indeed, but there's no _final_ cubical coalgebra. What I don't have is a category with an associative bifunctor with a final cubical coalgebra but no final square coalgebra. On a _discrete_ category being the unique coalgebra is necessary and sufficient for being a final coalgebra. It's easy to see that in a semigroup if x is a unique solution to xxx = x then x is the unique solution to xx = x. After writing all that, I note that in Vaughan's example, * is not a functor. If it were, there would have to map from 1*1 to 1*3 and 1*1 would be a square coalgebra.