I was playing a bit the 2-category Cat trying to have a better understanding of the notion of a 2-category (strict I guess). The usual definition of a category that I use assumes that $Hom(A,B)$ is a set.
What is an analogue of that condition in 2-categories? I guess you need to have some size restrictions in order to have a higher-Yoneda. I think the class of natural transformations between two functors is not a set in general, am I right?