Certainly video and likely Internet distribution.
Cat 4? thats too bad. Cat 5 minimum is recommended. 20mhz vs. 100mhz of transfer. No big deal for Internet surfing. But streaming media will overconsume the bandwidth.
Any cables labeled? If not, thats step 1 to verify and label. You could put an HDMI source (blu ray player et all) and plug in a HDMI TV to determine which cable goes where.
You could use a tone generator on the HDMI / Cat 4 cables. Home Depot / Lowes available.