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.