and there is a topological connection too

[Francois Lamarche <Francois.Lamarche@loria.fr>]

Various replies to my query during the day (my thanks to Ronnie Brown
and Lutz Schroeder) made me realize that my SMC structure was actually
the lifting of the CC structure on reflexive graphs (those with a choice
of loop) to ordinary non-reflexive graphs. Here is a bit more detail:

We all know these two categories are categories of presheaves. So let G
and R be the categories such that Set^G is graphs and  Set^R is
reflexive graphs. There is an embedding  G --> R, which generates the
usual triple of functors between the presheaf categories. So it seems
this functorial machinery allows to transform the CC structure in Set^R
into an SMC structure in Set^G, preserving the forgetful functor. There
must be general conditions that allow this.

I'm not pursuing this any more right now, because it has to have been
done before, and in a much more general setting.

Now Michael's comment is also (among other things) about the tension
between graphs and reflexive graphs, and naturally there is Lawvere's
"Qualitative distinctions between some toposes of generalized graphs"
that says a lot about that tension.

there may be more in there than we suspect

Francois