Dualities in the theory of accessible categories

Through the notion of weakly sound class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel-Ulmer duality for locally finitely presentable categories, Diers duality for locally finitely multipresentable categories, and the Makkai-Paré duality for finitely accessible categories. In doing so, we extend these to the enriched setting, provide a more formal and unifying approach to the theory, and also discuss new dualities that arise as a consequence of our main theorem.