In this paper, we de ne the category PosAct-S of all poset acts; an S-act in the category Pos, with action preserving monotone maps between them. Also, we study the existence of the free and cofree objects in the categories PosAct-S. More precisely, we consider all forgetful functors between these categories and the categories Act-S of S-acts, Pos of posets, and Set of sets, and study the existence of their left and right adjoints.
