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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.You can assume (if you like) that $F$

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?
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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 initial algebra terminal coalgebra $\mu \nu F$, to find another $F' \colon \mathrm{Set} \to \mathrm{Set}$ whose terminal coalgebra initial algebra $\nu \mu F'$ is isomorphic to $\mu \nu F$. You can assume (if you like) that $F$ is the structure functor of an algebraic signature.

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?
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A functor whose initial algebra is another's terminal coalgebra

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 initial algebra $\mu F$, to find another $F' \colon \mathrm{Set} \to \mathrm{Set}$ whose terminal coalgebra $\nu F'$ is isomorphic to $\mu F$. You can assume (if you like) that $F$ is the structure functor of an algebraic signature.

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?