@Owen – yeah, that's pretty much as far as I got.

Specifically, we can pick \\(p\in M\\) and \\(q\in N\\) such that \\(\phi(1_N) = p\\) and \\(\phi(q) = 1_M\\).

We then get \\(\phi = n \mapsto G(n)\circ p\\) and \\(\phi^{-1} = m \mapsto q\circ F(m)\\).

This implies that \\(F\\) and \\(G\\) are both injective, also that \\(m \mapsto m\circ p\\) and \\(n \mapsto q\circ n\\) are both surjective.

But I'm not sure there's much more we can say.

Specifically, we can pick \\(p\in M\\) and \\(q\in N\\) such that \\(\phi(1_N) = p\\) and \\(\phi(q) = 1_M\\).

We then get \\(\phi = n \mapsto G(n)\circ p\\) and \\(\phi^{-1} = m \mapsto q\circ F(m)\\).

This implies that \\(F\\) and \\(G\\) are both injective, also that \\(m \mapsto m\circ p\\) and \\(n \mapsto q\circ n\\) are both surjective.

But I'm not sure there's much more we can say.