from what I can make out, if \$$\phi\$$ is our bijection from \$$N\$$ to \$$M\$$, the naturality condition amounts to saying that for all \$$m\in M\$$ and all \$$y, n\in N\$$ we have \$$\phi(n\circ y\circ F(m)) = G(n)\circ\phi(y)\circ m\$$ – but I'm not sure what that entails!