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The main goal of the paper is a description of connected and projective objects of classes of categories that include categories of acts along with categories of pointed acts. In order to establish a general context and to unify the approach to both of the categories of acts, the notion of a concrete category with a unique decomposition of objects is introduced and studied. Although these categories are not extensive in general, it is proved in the paper that they satisfy a version of extensivity which ensures that every noninitial object is uniquely decomposable into indecomposable objects.