Incidence Category of the Young Lattice, Injections Between Finite Sets, and Koszulity

We study the quadratic quotients of incidence category Young lattice defined by zero relations corresponding to adding two boxes same row, or column, both. show that last quotient corresponds Koszul dual original category, while first are, in a natural way, duals each other and hence they are particular self-dual. Both these known be basic representatives Morita equivalence class injections between finite sets. also present new, rather direct, argument establishing this equivalence.