@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.