category of fraction and set-theoretic problem

[Philippe Gaucher <gaucher@irmasrv1.u-strasbg.fr>]

Bonjour,


I have a general question about localizations.

I know that for any category C, if S is a set of morphisms, then 
C[S^-1] exists. And moreover if C is small, then C[S^{-1}] is small
as well (as in the Borceux's book Handbook of categorical algebra I)

If S is not small, and if we suppose that all sets are in some universe
U, then the previous construction  gives a solution as a V-small category
for some universe V with U \in V (the objects are the same but the homsets
need not to be U-small). So it does not work if one wants to get U-small 
homsets.

Another way is to have a calculus of fractions (left or right) and if
S is locally small as defined in Weibel's book "Introduction to homological
algebra". 

But in my case, the Ore condition is not satisfied. Hence the question : 
is there other constructions for C[S^{-1}] ?


Thanks in advance. pg.