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.