\\(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.)

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.)