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.