We consider three aspects of homomorphisms of graphs and hypergraphs which are related to the structure of colour classes: (1) density, (2) the fractal property and (3) the generation of colour classes. In particular, we prove a density theorem for hypergraphs and show that, for connected oriented graphs, all jumps are balanced (and give an example to show that connectivity is needed here). We also show that a Hajós-type theorem holds for any colour class of undirected graphs, thus providing further evidence of the well-known ‘non-effective’ character of Hajós' theorem.