Perhaps an ill posed question, but here it goes anyway. Does the notion of adjunction in the case of monoids reduce to something formerly known? I mean: When the categories are preorders, an adjoint situation is a Galois connection. In the case when one monoid is homomorphic to another, is there a best-effort inverse, "from above and below", given by a construction named X, that complies with the definition of adjunction for single-object categories?