Injective and projective poset acts

In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in Pos} posets, with action preserving monotone maps between them, some categorical properties are considered. particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers etc. category. Also, several kinds epimorphisms monomorphisms characterized PosAct}-$S$. Finally, study injectivity projectivity respect to (regular) epimorphisms, respectively, see that although there is no non-trivial injective act monomorphisms, has enough regular injectives monomorphisms. it proved exactly retracts cofree complete posets.