The underlying functor : cat ----> graph, from small categories to small graphs, is tripleable. I would like to consider the same question over an arbitrary topos, and would appreciate any hints from topos theorists. It seems to me that, given a topos E, the underlying functor: cat(E) ----> graph(E) is "equationally defineable", that is to say, each internal category in E can be described by (in fact finitary) operations on some object of graph(E), subject to some equations. Then cat(E) ----> graph(E) is tripleable if and only if it has a left adjoint. It seems that constructing free categories over graphs requires some kind of countable colimits, which I don't imagine are always available. Perhaps the existence of a natural numbers object (which seems to be in some sense the most primordial of infinite colimits that a topos might possess) ensures the existence of free categories. Can such toposes with free categories be characterized? Thank you, Paul. =================================