The Galois connection between syntax and semantics

In his classic Dialectica paper `Adjointness in foundations' (1969), F. William Lawvere writes of `the familiar Galois connection between sets of axioms and classes of models, for a fixed [signature:]'.

But even if long familiar folklore to category theorists, the idea doesn't in fact seem to be that widely known. The ideas however are pretty enough, elementary enough, and illuminating enough to be worth rehearsing briskly in an accessible stand-alone form. Here's my attempt.

I wrote these...