The Myhill isomorphism theorem does not generalize much

The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduce two subsets of $\mathbb{N}$ to one another, there exists a bijection $\mathbb{N} \to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to whic