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
embeddings 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 et al., who
used it to obtain information about regular injectivity