If you transfer his iTunes library to your computer you do.
What happens if there is a problem with the update? How will he be able to restore his iPhone as is including all iTunes content and any photos transferred from his computer if you don't transfer this data from his computer to your computer?
What is the problem with authorizing your computer with his iTunes account under his computer login - for an iTunes account that is already set up - no set up required?
If you don't have enough available storage space to back up his iPhone, you should be moving files off your Mac's hard drive anyway. You can always place his iTunes library on an external drive for him to access his library from his computer login account on your computer.