Okay, let's do it, Jesus! Say we have monoids \$$M\$$ and \$$N\$$, which we think of as categories, each with one object which I'll call \$$\star\$$... we're smart enough to keep track of when we're talking about \$$\star_M\$$ and \$$\star_N\$$, I hope. Let's figure out what adjoint functors

\$F: M \to N, \quad G: N \to M \$

amount to!

First of all, can you (or anyone) guess what a functor \$$F: M \to N\$$ amounts to in this case? It's something familiar from the study of monoids.