Syntax is NOT the adjoint of semantics.
The term ‘syntax’ refers not to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent. 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. Happy New Year! Bill