You can do hdmi over cat5/6 with something similar to this
Cable Matters Wall Mount HDMI Extender Balun (HDMI Over Ethernet Cable/HDMI Over Cat6 Cable) with TCP/IP Support for 1-to-Many Setup - Up to 300 Feet https://www.amazon.com/dp/B00OZV04BK/ref=cm_sw_r_cp_apa_i_t9peDbPCNDGZR