Clarification of definition of category Need the set of objects and the set of non-id morphisms be disjoint? If not, can the morphisms be a subset of the objects?
 A: Disjointness is a red herring: there is never a situation where objects and morphisms are considered on equal grounds. However, it is often convenient to assume that the two are disjoint.
There is always an injection of the set of objects into the set of morphisms, namely the map $x \mapsto \mathrm{id}_x$, so the cardinality of the set of morphisms is always at least the cardinality of the set of objects. 
A: First of all, neither the objects nor the morphisms are required to be a set, they can be any class, even a proper one. Second, the definition does not forbid the morphisms to be a subclass of the objects. I can't think of any non-artificial example, though.
A: In the most direct (set theoretic) construction of $\mathbf{Set}$, the category of small sets, the objects are small sets, and the morphisms are set-theoretic functions. Because every function between small sets is itself a small set, the morphisms of $\mathbf{Set}$ are indeed a subclass of the objects of $\mathbf{Set}$.
Depending on your precise definition of category, you might instead need the morphisms to be triples $(A,f,B)$ where $A,B$ are sets and $f$ is a function whose domain is $A$ and whose range is a subset of $B$. But the usual set-theoretic definition of triples make these into small sets, so we still get the same result.
A: I suppose one situation in which you might care is if you want to think of a functor $f:C\to D$ as a monolithic function that maps $C$-data to $D$-data. Then you would want the set of $C$-data to be a disjoint union of the object set of $C$ and all its homsets. But I don't know anyone actually takes this viewpoint. (And as paul garrett  points out, for non-small categories, these need not be sets.)
