I'm basing myself on the Wikipedia article on Categories. A category $C$ consists on:
- a class of objects $Ob(C)$
- a class of morphism $Mor(C)$
- a class function going from $Mor(C)$ to $Ob(C)$ called domain
- a class function going from $Mor(C)$ to $Ob(C)$ called codomain
- etc
While I don't totally understand the concept of classes (and hence, of class functions), I am yet faced with this problem: 'domain' and 'codomain' being functions, they must be morphisms. Hence the have a domain and a codomain. But the domain and codomain of these domains and codomains must be again morphisms ; and this goes on and on ?