Collapsing objects in a category Suppose I have a category C. And there are two objects X and Y with no morphisms between them. I've checked up "quotient category" on wikipedia, but there I can only make isomorphic objects with morphisms between them.
Is there a categorial notion available that I can use in this case? 
 A: Short answer: yes. There is a special class of 2-categorical limits called iso-inserters that does the trick. The paper to check is M. Kelly's "Elementary observations on 2-categorical limits", Bull. Austr. Math. Soc. 39 (1989), 301-317.
Instead of explaining what these are let me go about explaining how you would make two objects $a$ and $b$ isomorphic. First pass to the underlying graph of the category and insert two edges $a\to b$ and $b\to a$ then take the free category of this new graph. You have a graph morphism from the original category into this new graph. Quotient the category to force the graph morphism to be a functor. Now quotient the category again to make the edges you inserted to be mutually inverse. There is only a small snag to this construction: the category you end up may not be locally small, because inserting isomorphisms may create a proper class of new morphisms. This is where things like "calculus of fractions" come in.
Hope it helps, regards, G. Rodrigues
A: Domenicos comment lead to the following idea (I am posting this as an answer, as it is too long for a comment):
Let $CAT$ denote the category of small categories and $CAT'$ denote the category, whose objects are small categories except for the fact that the composition needn't be defined on the whole of $Mor(A,B)\times Mor(B,C)$ (but just on a subset of it). Associativity and so on should hold, whenever it is defined. 
Then there is a obvious inclusion Functor $CAT\rightarrow CAT'$. One should check, whether it has a left adjoint $L:CAT'\rightarrow CAT$. Then one can make out of the data above a object in CAT' by adding an additional isomorphism from $X$ to $Y$ and one doesn't have to worry aboutthe compositions of that iso with the morphisms in CAT. Using $L$ one could make a honest category out of this.
