Edit: the question was unclear, so prompted by the comments and answers I've tried to clarify things.

A colleague has asked me whether, or under what conditions, it is possible, given a sufficiently nice functor $F \colon \mathrm{Set} \to \mathrm{Set}$ with terminal coalgebra $\nu F$, to find another $F' \colon \mathrm{Set} \to \mathrm{Set}$ whose initial algebra $\mu F'$ is isomorphic to $\nu F$, with the two structure maps being inverses up to this isomorphism.

In the case m'colleague is considering, F is the structure functor of an algebraic signature and $\nu F$ is the set of possibly infinite terms coinductively generated from it. Apparently the same set of terms can be generated inductively from a strictly larger signature, whose structure functor would then have an initial algebra, and he would like to know if there is a more abstract category-theoretic perspective on this situation. (Note that F and F' will in general not be the same.)

It's an interesting question, and I'd like to help, but I'll be horribly busy for the next while and won't have time. Any ideas on where he could start looking?

slaps foreheadI got mixed up: he has the terminal coalgebra of one functor and wants it to be the initial algebra of another. The first is the set of infinite terms coinductively generated from a signature. He has reason to believe that the same terms can be generated inductively from an enlarged signature, and wants to know if there is a more abstract category-theoretic perspective on this. – Finn Lawler Feb 21 '11 at 21:06structure mapsof the terminal coalgebra and the initial algebra. – Andrej Bauer Feb 22 '11 at 11:00