> 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?

I don't think this is an ill-posed question.

In the event of a lattice equipped with a monoid with two adjoints for \\(\bullet\\), we call such a structure a [*residuated lattice*](https://en.wikipedia.org/wiki/Residuated_lattice#Definition).

I believe another example of a residuated lattice is a [quantale](https://en.wikipedia.org/wiki/Quantale).