"There is very little that Barnard offers to any Columbia student."
Well, money. Barnard doesn't just mooch off the university; they pay exorbitant fees to the Trustees of Columbia for use of facilities. And, of course, Columbia students have access to Barnard professors, many of whom (depending on the department) are in the top of their field. Obviously, Columbia University isn't dependent on Barnard—they have 3 undergrad colleges, 20 grad schools, and an $8 billion endowment—but it's not as though they don't derive any benefit from the inter-corporate agreement. I, for one, think my experience at Columbia would have been much worse had Barnard not existed.