Category theory, and for that matter modern (as opposed to elementary) algebra, is to mathematics as mathematics is to physics, and for that matter to computer science. Whereas mathematics organizes reasoning about the phenomena studied by physicists and computer scientists, algebra and category theory perform a similar function for mathematics.
In any setting organization is desirable, and arguably necessary on occasion. But the use of algebra and category theory to organize physics and computer science is a double whammy here. One should therefore be doubly sympathetic of those physicists and computer scientists who want to know what substantive contribution is being made to their subject and can't evaluate the answers because they are one if not two levels removed from the necessary abstractions.
Vaughan Pratt
It just occurred to me that to justify such viewpoint we might have to look at the point in time when mathematics began to become abstracted out from natural sciences to see whether category theory is already in the same position with respect to the rest of mathematics. Although I certainly do not know enough history of science, I will still dare to speculate that the situation now is completely different from what it was then. I believe mathematics as a substantial part of the body of scientific knowledge did exist and evolve long long before it began to be considered as some separate entity which can be used to organize the rest - in fact many people still think of mathematics as just another science on completely equal footing with, say, biology or physics. Whereas birth and development of category theory has been, I think, much more deliberate, abrupt and discontinuous in comparison. If this is so, one possible conclusion might be that probably category theorists simply want too much too soon. Maybe they should be more patient and let their discipline become stronger within the body of mathematics before forcibly declaring it a new organizing force outside the rest of mathematics. This is as if a child would be forced to care for its parents shortly after being born. Mamuka Jibladze