 The purpose of this section is to compare FA and AFA in a  technical way, using ideas from category theory.    That is, the  language of category theory and especially its  built-in feature of duality are used to say something insightful about the relation between FA and AFA.    Further, the dual statements about the axioms  suggest a much more systematic and thoroughgoing  duality about a host of other concepts.    This deeper point is not a strictly mathematical result but rather more of a research program, and so the final subsection here will detail some of what is known about it. 