In response to Ph. Gaucher's question: I (try to, at least...) treat this question for weak $n$-categories in my preprint ``Limits in $n$-categories'', available on the xxx preprint server as alg-geom 9708010. If I understand correctly, the set-theoretical problem you raise is the same as the one encountered in section 5 of my preprint. The conclusion is that the (weak) $n+1$-category $nCAT$ is closed under direct limits. It seems that coproducts of strict $n$-categories, if they exist, cannot actually be the ``right'' ones because in that case, every weak $n$-category would be equivalent to a strict one. I haven't made this argument rigorous, though. ---Carlos Simpson PS what is a ``comma category'' or ``comma object''?
The argument sounds reasonable. Before this question, I was convinced of the existence of this cogenerator. I have to find something else for the lemma I would like to prove...
Since it does not exist, I have another questions (I suppose well- known) and any reference abou the subject would be welcome :
How does one prove the cocompleteness of omegaCat (small & strict) ? The only idea of proof I had in mind until this question was : omegaCat is obviously complete (and the forgetful functor towards the category of Sets preserves projective limits), and well-powered and a cogenerator => the cocompleteness (Borceux I, prop 3.3.8 p 112).
Without cogenerator, how can one prove the cocompleteness ? The explicit construction of the colimit seems to be very hard : the forgetful functor towards Set does not preserve colimits because the underlying set of the colimit might be bigger than the colimit of the underlying sets. Every time two n-morphisms are identified in the colimit of the underlying sets, p-morphisms (with p>n) might be "created" by the colimit.
Thanks in advance for any answer. pg.