Well, to start off, you have already finished your OSCP. However, I'm going to assume the reason why your professor asked you to pentest the University's Network is as a "final project" in which you make use of what you know. Saying that your interest is to test what you've learnt is a good idea.
Now, answering your question, Yes, you should present paperwork, but usually written evidence from the organization's authorities will do in most cases. An E-Mail from your professor, another one from the University's Authorities, and, if possible, another one from the IT department would usually be enough.
However, since Penetration Testing can be carried with bad intentions, I would advice you to talk to your teacher and ask him to not only e-mail you, but also to e-mail the University's Institution and the IT asking for permission.
I would say that written evidence is a must when talking about carrying Penetration Testing. I still recommend you the contract option, it may seem overkill, but it is, as I have said, the safest option. If I were you, I would get both a contract, and other evidence like e-mails, etc.
Hope it was helpful.