Michael Barr recalls the definition of a category object via category structures on the Hom(-,c), which does not necessitate pullbacks. It was given by Grothendieck under the name of "structure de categorie sur l'objet c " in "Techniques de descente...", Seminaire Bourbaki, 195, 1960. However, as much as I know, he did not consider the case of differentiable categories which Charles Ehresmann had introduced earlier. A similar Hom based definition for groups was studied by Eckmann and Hilton in a 3 parts paper "Group-like structures in general categories", published in Math. Annalen 145 (1962) and 151 (1963). In his paper "Introduction to structured categories" (Technical repot 10, Univ. Kansas 1966; reprinted in the "Oeuvres" part III-2), Charles compares the Grothendieck notion with his 1963 notion of structured category (or internal category in a concrete category) and with the general notion of internal category defined in this 1966 paper of as a model of the sketch of categories. Sincerely Andree 14-May-2002 13:52:00 -0300,1307;000000000000-00000000