Hi, Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a "V-class" is a subset of V. Say that a category C is "V-included" when it has the following two properties. (1) ob C is a V-class. (2) C(x,y) is a V-set for all x,y in ob C. The advantage of V-inclusion over local V-smallness (i.e. condition (2) alone) is that V-included categories are W-small for every universe W greater than V, whereas locally V-small categories are not, in general. Furthermore, all the standard categories constructed from V are V-included.?? (Except for the ones that are not even locally V-small, like the category of V-included categories.) Is there a standard name for V-inclusion? Paul [For admin and other information see: http://www.mta.ca/~cat-dist/ ]