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