Keith and Valter - yes, a functor between one-object categories is the same as a monoid homomorphism!
So, when we have two functors, a left adjoint \\(F : M \to N\\) and a right adjoint \\(G: N \to N\\), between one-object categories, we should think of these as monoid homomorphisms.
But then there's more! We need a natural one-to-one correspondence between morphisms
\[ f: F(m) \to n \]
and morphisms
\[ g: m \to G(n) \]
for every pair of objects \\(m \in \mathbf{Ob}(M), n \in \mathbf{Ob}(N).\\) That's what makes \\(F\\) and \\(G\\) adjoints.
But this requirement can be simplified, since we're dealing with monoids. How?
(At first let's not worry about the _naturality_ requirement; that's very important, but it can be put off to the end.)