  The purpose of this paper is to show how Positselski's relative
nonhomogeneous Koszul duality theory applies when studying the linear category
underlying the PROP associated to a (non-augmented) operad of a certain form,
in particular assuming that the reduced part of the operad is binary quadratic.
In this case, the linear category has both a left augmentation and a right
augmentation (corresponding to different units), using Positselski's
terminology.
  The general theory provides two associated linear differential graded (DG)
categories; indeed, in this framework, one can work entirely within the DG
realm, as opposed to the curved setting required for Positselski's general
theory. Moreover, DG modules over DG categories are related by adjunctions.
  When the reduced part of the operad is Koszul (working over a field of
characteristic zero), the relative Koszul duality theory shows that there is a
Koszul-type equivalence between the appropriate homotopy categories of DG
modules. This gives a form of Koszul duality relationship between the above DG
categories.
  This is illustrated by the case of the operad encoding unital, commutative
associative algebras, extending the classical Koszul duality between
commutative associative algebras and Lie algebras. In this case, the associated
linear category is the linearization of the category of finite sets and all
maps. The relative nonhomogeneous Koszul duality theory relates its derived
category to the respective homotopy categories of modules over two explicit
linear DG categories.
