Addendum

Sorry for the clutter. I forgot a counterexample. Yes, it is possible for (X*X)*X to have a final coalgebra but not X*X. On the discrete category with two objects, A and B, let the bifunctor be defined by: * | A B --+------ A | B A B | A A The unique cubical coalgebra is A but there is no square coalgebra. What I don't have is a counterexample with an _associative_ bifunctor.