  In this paper, the categorial property of compactness of an object, i. e.
commuting of the corresponding $\Hom$ functor with coproducts, is studied in
categories of $S$-acts and the corresponding structural properties of compact
$S$-acts are shown. In order to establish a general context and to unify the
approach to both of the most important categories of $S$-acts, the notion of a
concrete category with unique decomposition of objects is introduced and
studied.
