Let M be a class of (mono)morphisms in a category A. To study mathematical
notions, such as injectivity, tensor products, flatness, one needs to have some categorical and
algebraic information about the pair (A,M).
In this paper, we take A to be the category Pos-S of S-posets over a posemigroup S, and
Mdc to be the class of down closed monomorphisms and study the categorical properties, such
as limits and colimits, of the pair (A,M). Injectivity with respect to this class of
monomorphisms have been studied by Shahbaz and Mahmoudi who used it to obtain
information about regular injectivity.