Given the set 𝒮N of all operator endomorphisms T with the property that whenever D is any 2-density operator, then T (D) is N-representable is considered. We specify a pair {Q2,PN} of extreme points of 𝒮N which are related to each other via the particle-hole duality (Theorem 1.2). The rest of the paper is devoted to a complete characterization of the set ŜN of all operator endomorphisms T with the property that whenever D is any 2-density operator then T (D) is quasi-N-representable, i.e., satisfies the necessary conditions Id, QN, BN, and CN.