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.)

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.)