How to prove νX. A × X ≅ (μX. 1 + X) -> A?

How can we prove Stream A = νX. A × X is isomorphic to Nat -> A = (μX. 1 + X) -> A ?

In programming sense, Stream A can be seen as a function from Nat to A, and I can write isomorphisms between them. But how can this be proven mathematically?

I would also like to know the conversion mechanism from μ to ν, and vice versa.