Frequently, mathematical structures of a certain type and their morphisms fail to form a category for lack of composability of the morphisms; one example of this problem is the class of probabilistic automata when equipped with morphisms that allow restriction as well as relabelling. The proper mathematical framework for this situation is provided by a generalisation of category theory in the shape of the so-called precategories, which are introduced and studied in this paper. In particular, notions of adjointness, weak adjointness and partial adjointness for precategories are presented and justified in detail. This makes it possible to use universal properties as characterisations of well-known basic constructions in the theory of (generative) probabilistic automata: we show that accessible automata and decision trees, respectively, form coreflective subprecategories of the precategory of probabilistic automata. Moreover, the aggregation of two automata is identified as a partial product, whereas restriction and interconnection of automata are recognised as Cartesian lifts.
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