Hello, let me add some remarks to Marco Grandis' posting.
1. Generalised quotients of categories.
A very general notion of generalised congruence in a category - also involving objects - can be found in a paper by Bednarczyk, Borzyszkowski and Pawlowski [BBP].
I have not yet looked at that paper, but I think the "natural" thing is to consider equivalence relations R on a category C, which are subcategories of CxC (i.e. closed under composition, not necessary full; identities are in R by relexivity of R). In my diplomarbeit "Kongruenzrelationen auf Kategorien" from 1977, I considered that, but I was not the first one. Some years earlier there was a paper by Jacques Mersch from Liege (Belgium), which unfortunately appeared only in an internal publication of the university of Liege. Moreover, I think I remember that John Isbell did something on this subject.
The quotient p: X -> X/A is determined by the obvious universal property:
The universal property is also obvious in the general situation. For small categories, a functor with this property always exists, let's call it a quotient functor. For large categories it my happen that the hom-sets of the quotient become large, even if the hom-sets of the original category are small. In general, a congruence as above is not a kernel of a functor; the quotient functor may identify more morphisms. In my diplomarbeit, I rediscovered an example, which had already been found by Mersch. The quotient functors are exactly the regular epis in CAT. But unfortunately, they are not closed under compositon, so a quotient of a quotient of C need not be a quotient of C.
It is interesting to note that p automatically satisfies a 2- dimensional universal property, as one can easily deduce from the fact that natural transformations can be viewed as functors X -> Y^2, with values in the category of morphisms of Y.
Of course, this argument also works in the general situation. Greetings Reinhard Boerger