2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229291We review Koszul duality in representation theory of category $ \cal O $, especially we give a new presentation of the Koszul duality functor. Combining this with work of Backelin, we show that the translation and Zuckerman functors are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel and Khovanov. Finally we use Koszul duality to give a short proof of the Enright-Shelton equivalence.Published version, Journal of Lie TheoryRepresentation Theory16S37text