\$$M\$$ and \$$N\$$ are one-object categories, so we must have \$$m = {\star}_M\$$ and \$$n = {\star}_N\$$

Hence we need a bijection between morphisms \$$f : F({\star}_M) \rightarrow {\star}_N\$$ and morphisms \$$g : {\star}_M \rightarrow G({\star}_N)\$$

ie, a bijection between elements of \$$N\$$ and elements of \$$M\$$

(Now we just have to worry about naturality, at which point I get kinda stuck. I think I can prove that if this bijection is a homomorphism, then it equals \$$g\$$, and \$$f = g^{-1}\$$. But I don't believe we can rule out the case where the bijection _isn't_ a homomorphism.)