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.
