Al Vilcius asked:

If F:C-->C is a functor with opposite F*:C*-->C>* then (F-algebras)* is trivially equivalent to (F*)-coalgebras. Can this duality be induced by a schizophrenic object?

The answer is, in general, negative: every duality, where C has a terminal object, is of the above type, just take the constant functor of value 1. J. Adamek