TITLE: Why need the morphisms to form a set ?  
QUESTION [7 upvotes]: For a category $C$ it is required that the morphisms of any two objects of $C$ form a set (c.f. Lang: Algebra, or Weibel: An introduction to homological algebra). 
What's the point about this requirement ? Would there be any disadvantages / logical deficiencies if one allows the morphisms to form a proper class ?

REPLY [4 votes]: As David mentioned, category theorists generally tend to shy away from such `size issues', because in a sense they do not touch the heart of the matter, as the original question rightfully suggests. Apart from such foundational issues, I can think of two practical reasons where it is important that homs be sets.
First, there is Freyd's celebrated Adjoint functor theorem. It gives conditions that characterize precisely when a given functor has an adjoint. Crucially, one of the conditions, called the `solution set condition', is that a certain class is in fact a set. This shows that size issues do play a fundamental role in category theory, which came as quite a surprise to most people.
Second, one can think of Enriched categories. It turns out that a lot of category theory goes through if homs are not necessarily objects of the category Set of sets and functions, but objects in an arbitrary monoidal category, with composition being a morphism of that category. For example, relating to Kevin and Qiaochu's comments above, a 2-category can be seen as a category enriched in Cat, the category of categories. But this also gives some surprising examples. For example, a metric space can be seen as a category enriched in $\mathbb{R}^+$, i.e. the poset $[0,\infty]$ with monoidal structure given by addition. And of course a locally small category is just a Set-enriched category. This is not an argument against `large categories' per se, but does indicate that a lot of murky waters can be avoided by only considering locally small categories.