Let's start at the beginning.
Your cable TV is ,for all intents and purposes, some sort of radio frequency signal,, be it digital or analog. It doesn't matter.
This is a totally different creature from a video signal, be it composite of component.
To convert that RF signal to a video signal, you need some sort of device, which type is yet to be determined. A simple cable with different ends won't do it. That would be like connecting an FM antenna to the input of an audio amplifier and expecting sound. ...ain't gonna happen.
Now, if the university sends out an unencrypted analog signal, any functioning VCR or TV will work. It won't be hi def but, hey, who cares?
If they send out an unencrypted digital signal, any TV or VCR with a QAM* tuner will work. I can't recall any VCR's with that, but I could be wrong.
If they send out an encrypted signal. either analog or digital, you're probably gonna need some sort of set-top box in order to get a picture.
'effens I were you, I'd check with the powers that be to see exactly what you need to do what you want to do. Once you find that out, perhaps we can provide some guidance instead of playing battleship.
And, no, composite and component are not interchangeable on a switch.
* QAM = digital cable TV tuner. ATSC = digital OTA (Over The Air, or antenna) tuner