Dualities for accessible categories

In this work we prove dualities for Diers categories, Barr categories and small Barr-exact categories. The latter duality solves a problem of M. Makkai. Further, we prove a stronger version of the strong completeness theorem on $ kappa$-Barr-exact categories. Finally we prove that an accessibly embedded subcategory of a locally presentable category satisfies the solution-set condition if it is accessible. This improves work of J. Adamek and J. Rosicky on injectivity in locally presentable categories