Importance of local smallness when checking functor diagrams? I have a question concerning the importance or non-importance of assuming local smallness for a category $\mathcal{C}$. 
There are two procedures I have done while working through an exercise that have given me a bit of doubt. First, I had to define a functor out of a particular product category. The arrows in this category fall into three different types, so I specified what my new functor does to each type of arrow, and then verified that identities and compositions were preserved. This feels an awful lot like defining a function on a set, so when I do this, am I implicitly assuming that my category is locally small (or even small?). I'm not sure that I am, and the exercise doesn't mention local smallness (even though elsewhere in the book the author does mention local smallness in certain instances, so I don't think this was an oversight). 
At another stage in the exercise, I had to check that a diagram of functors commuted. To check the objects I traced an arbitrary object through, and to check the arrows I traced an arbitrary arrow through. Does anything about this procedure also implicitly assume local smallness or smallness of $\mathcal{C}$? Since it feels like I am tracing an element of a set? Or does this avoid foundational issues like whether or not the maps fall into a proper set or a class? Or is it just that there are some aspects of working with classes that are like working with a set in practice?
 A: The short answer is, no, you're not assuming local smallness in such argument.
The long answer is that even asking this question requires some understanding of what a category which isn't locally small means. What are the hom-objects? They're "classes" of some kind, but you have to formalize what a "class" is. The most conservative approach to this is NBG set theory. However, NBG is a bit inconvenient in that it for instance cannot prove the existence of categories of functors between non-small categories. The most popular foundation among many mathematicians who use category theory is the axiom of Grothendieck universes, which says in effect that every category is small in a sufficiently large set-theoretic universe. 
Thus all concerns like yours totally dissolve, which is the correct thing to have happen if your goal is really to learn category theory, while you should focus more carefully on these things if you hope to do research in the interaction between set theory and category theory.
