  By applying simplification operations to categories of multigraphs, several
natural graph operations are shown to demonstrate categorical issues. The
replacement of an undirected edge with a directed cycle for digraphs admits
both a left and a right adjoint, while the analogous operation for quivers only
admits a left adjoint. The clique-replacement graph, intersection graph, and
dual hypergraph fail to be functorial with traditional graph homomorphisms. The
three failures are remedied by considering weak set-system homomorphisms, which
form a category isomorphic to both the category of incidence structures and a
lax comma category.
