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Mike Shulman wrote a nice expository paper on set theoretical foundations for category theory

http://arxiv.org/abs/0810.1279

In Section 6 he explains the difficulties of working with large categories using just ZFC, and he discusses various ways to deal with these size issues. Some of these do not assume the existence of an inaccessible cardinal.

show/hide this revision's text 1

Mike Shulman wrote a nice expository paper on set theoretical foundations for category theory

http://arxiv.org/abs/0810.1279

In Section 6 he explains the difficulties of working with large categories using just ZFC, and he discusses various ways to deal with these size issues.