In Steve Awodey's book on category theory, he claims the latter is a branch of abstract algebra. I've never seen such a classification before. Is this really correct?
One could likewise argue that General Algebra (which encompasses groups, rings, lattices, and many other structures) is the study of a special class of categories, and so argue that General Algebra is a branch of Category Theory... (For example, George Bergman's General Algebra book is heavily category-theoretically flavored).
I would say rather that Category Theory has some very large areas of intersection with General/Abstract Algebra; I remember George Bergman saying once that in the 80s (?) there was a big conference at Berkeley/MSRI that invited both Universal Algebraists and Category Theorists, and that many times over the course of the conference they discovered that there were results that each "camp" had proven independently and did not realize the other "side" knew about them, or that there were questions that had been raised on the periphery of one whose answer was well-known by the other. (I hope I'm not misremembering and/or misreporting this!).
My particular (heavily algebra-biased) experience leads me to think that Category Theory is closer to abstract algebra than to other branches of pure mathematics (e.g., topology, analysis, etc). Don't know if I would go so far as to call it a "branch" of abstract algebra, though, any more than I would call it a branch of set theory (or set theory a branch of category theory).