Semantics-structure adjunction

My comment "Daniel has come up with what seems to be the right answer to this question" was written on revision 8 in 2009. I think the answer I had in mind appears in Tannaka duality for comonoids in cosmoi (arXiv:0911.0977, hence Nov. 2009), specifically this passage from section 5:

These names go back to Lawvere (see [Law04, p. 77]). A monad can be viewed as a sort of logical theory, and from this viewpoint the semantics functor sends it to its category of models; the study of the models of a logical theory is generally called its semantics. On the other hand, a monad can also be seen as a ‘type of structure’ with which objects of $C$ can be equipped. From this point of view, the structure functor sends a category $A$ over $C$ to the universal structure with which the objects of $A$ can be equipped.

I am not sure about what prof. Shulman refers to, but I am pretty confident that the answer to your question is on page 74 of Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics 145, by Eduardo Dubuc:

There is an adjunction $\text{Str} \dashv \text{Sem}$,$$\text{Str}:\text{Cat}^*_{/A} \leftrightarrows \text{Monad}(A)^{\circ} : \text{Sem}$$ moreover $\text{Monad}(A)^{\circ}$ is reflective in $\text{Cat}^*_{/A}$ via this adjunction.

Sem maps a monad $T$ to the forgetful functor $U: \text{Alg}(T) \to A$, while Str maps a functor to its codensity monad. Observe that by $\text{Cat}^*_{/A}$ I mean the full subcategory of $\text{Cat}_{/A}$ of those functors that admit a codensity monad.

Of course, the same result can be obtained with density comonads and comonads.