water pump hose connector

[H2inMD]

Ok, its a 2003 H2. The hose from the water pump is leaking at the firewall

The other end of the hose goes into a black T connector near the water pump.

Always something to make it more difficult. What is this black connector, how do I connect the hose into this black t connector


And finally, I hope its just the hose and not the plastic thing it hooks to? I guess this is going to the Heater core???

help