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.