Extensions of the category $S-Act$

We define a new category $SS-Act$ whose objects are $S$-acts and whose morphisms are defined so that each set $Hom_{SS-Act}(A, B)$ is an $S$-act. It is proved that this category has a reflective subcategory $FS-Act$ that is naturally isomorphic to the category $S-Act$. The set $Hom_{FS-Act}(A,B)$ coincides with the set of all fixed points of the $S$-act $Hom_{SS-Act}(A,B)$. In the case when $S$ is a group, it is proved that the category $SS-Act$ is a Grothendieck topos and the construction of limits and colimits is considered.