Michael MAKKAI Title: Universal properties in the higher dimensional context Abstract Everyone knows that each universal property has two distinct, but equivalent, formulations; one that I call the "small" one, the other the "big" one. The small definition of (Cartesian) product is that of a product diagram: object A, B, AxB, and the two projections as usual; you have to give a *property* (the universal property) of these data to finish off the specification. The big definition of "product" consists of specifying the existence of a natural isormorphism hom(-,AxB)-->hom(-,A)xhom(-,B) ; this is big since a natural transformation of the said shape involves many more constituent parts than the product diagram. Of course, what connects the two is *Yoneda*. In this talk, I will explain how my recent definitions (available in written form through the usual channels) of "multitopic omega-categeory", and of MltOmegaCat, the "multitopic omega-category of all multitopic omega-categories", fit into the pattern of the "big" definitions of universal properties. In fact, this explanation can be given so that it leads to the (finally indispensable) technical definitions, rather than starting with those definitions, and trying to explain them. Let me note, in passing, that there are equivalent "small" versions of the definitions; and in fact, these were the ones I was working with in the numerous examples I gave in my talks in the spring term. In fact, the Baez/Dolan work (remember: I am standing on the shoulders of these individuals ...) is doing the thing in the "small" style. It seems, however, that in the present context, the "big" way is somehow "smaller" (more comprehensible) than the "small" one; this is not surprising, in view of our shared feeling that adjoint functors are the real way of talking about universal properties. All the above opens up the question of *Yoneda* in the multitopic context. I will say things about *Yoneda*. I will also say things about further universal properties, in particular, about the Cartesian closed nature of MltOmegaCat. These latter things are not in the paper I mentioned above. M. Makkai