  In the standard Category of Graphs, the graphs allow only one edge to be
incident to any two vertices, not necessarily distinct, and the graph morphisms
must map edges to edges and vertices to vertices while preserving incidence. We
refer to these graph morphisms as Strict Morphisms. We relax the condition on
the graphs allowing any number of edges to be incident to any two vertices, as
well as relaxing the condition on graph morphisms by allowing edges to be
mapped to vertices, provided that incidence is still preserved. We call this
broader graph category The Category of Conceptual Graphs, and define four other
graph categories created by combinations of restrictions of the graph morphisms
as well as restrictions on the allowed graphs. We investigate which Lawvere
axioms for the category of Sets and Functions apply to each of these Categories
of Graphs, as well as the other categorial constructions of free objects,
projective objects, generators, and their categorial duals.
