  We study the Koszul duality between augmented $E_n$-algebras and augmented
$E_n$-coalgebras in a symmetric monoidal stable infinity $1$-category equipped
with a filtration in a suitable sense. We obtain that the Koszul duality
constructions restrict to an equivalence between augmented algebras and
coalgebras which have some positivity and completeness with respect to the
filtration. We also obtain that the Koszul duality construction is functorial
between carefully constructed generalized Morita categories consisting of those
algebras/coalgebras in each dimension.
