Contravariant Koszul duality between non-positive and positive dg algebras

The Koszul dual of locally finite non-positive dg algebra is locally finite positive dg algebra. However, the Koszul dual of locally finite positive dg algebra is not necessary locally finite. We characterize locally finite positive dg algebras whose Koszul dual is locally finite. Moreover, we show that the Koszul dual functor induces contravariant equivalences between the perfect derived category and the perfectly valued derived category. As an application of Koszul dualities, we establish an ST-correspondence. We also show that, under some assumption, every covariantly finite bounded heart is a length heart, and the triangulated analogy of Smal{\o}'s symmetry holds.