In this paper, first the congruences in the category PosAct-S of all poset acts over a pomonoid S; an S-act in the category Pos of all posets, with action preserving monotone maps between them, are introduced. Then, we study the existence of the free and cofree objects in the category PosAct-S. More precisely, we consider all forgetful functors between this category and the categories Pos-S of all S-posets, Pos of all posets, Act-S of all S-acts, and Set of all sets, and we study the existence of their left and right adjoints. It is shown that the category Pos-S is a full reflective and coreflective subcategory of PosAct-S.
