Foundations in category theory I'm new in this study and I don't know much about the foundations of mathematics, so I have a question. 
If I'm doing category theory, and I need to talk about "small categories" , "locally small" and etc, I need to have some set theory foundation like NBG where I can talk about this things, right? But, if I want to take the category of all categories, I will have some problem in this context with NBG as foundation? 
 A: This may be a little deeper than you're looking for, but the papers numbered 58 and 62 on this list are ones that I found thought-provoking. Foundations of category theory is a subject that Feferman has returned to a few times, and he has the nice device of a recurring list of requirements for his ideal theory. He largely hovers near NFU-esque theories, but I think they should be of interest even if you're interested in more conventional foundations in the long run.

As for NBG, remember that we define categories as tuples, and generally as Kuratowski tuples. So with any collection of small categories, we can apply sumset a few times and an instance of separation to get, say, the collection of all the objects of those categories. So if we had a collection of all small categories, we could apply sumset/separation a few times to get the collection of all sets. In NBG or a similar theory, one needs to simply let some categories not be members of anything. In practice, this isn't such a big deal, and a few extensions of the theory exist that give us a bit more elbow room among the sets if needed.
