We give a unified proof of Gabriel-Ulmer duality for locally finitely presentable categories, Adámek-Lawvere-Rosick´y duality for varieties and Morita duality for presheaf categories. As an application, we compare presheaf categories and varieties.
Centazzo, C., & Vitale, E. (2002). A duality relative to a limit doctrine. Theory and Applications of Categories, 10, 486-497. https://hdl.handle.net/2078.5/152000 (Original work published 2002)