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!