I'm not sure I understand this
In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated).
Both of the functors
?--------------> theories -------------->Large categories Syntax Semantics
are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera.
Do you mean that if we choose some kind of sketches for the domain category then theories are a reflective subcategory, more or less the 'definitionally closed' sketches? Then a presentation of a theory T would be (up to isomorphism) any unit arrow of the adjunction with T as codomain? best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]