ct.category theory - Collapsing objects in a category

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 $CATrightarrow CAT'$. 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.

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