  We introduce the concept of a morphism between coloured nets. Our definition
generalizes Petris definition for ordinary nets. A morphism of coloured nets
maps the topological space of the underlying undirected net as well as the
kernel and cokernel of the incidence map. The kernel are flows along the
transition-bordered fibres of the morphism, the cokernel are classes of
markings of the place-bordered fibres. The attachment of bindings, colours,
flows and marking classes to a subnet is formalized by using concepts from
sheaf theory. A coloured net is a sheaf-cosheaf pair over a Petri space and a
morphism between coloured nets is a morphism between such pairs. Coloured nets
and their morphisms form a category. We prove the existence of a product in the
subcategory of sort-respecting morphisms. After introducing markings our
concepts generalize to coloured Petri nets.
