No. That is the hose from your stock bilge. You will eventuallt get rid of that plastic box and use the aluminum tube that it once used as your new inbound.
But you have to remove your pump, seal off the stock bilge suction thingy with some goop, and drill and tap a new hole for a new brass fitting. There is already a little spot on the pump ready for this. Then you will have true dual cooling.
Then you simply reconnect the hose at the pump which will now push instead of pull.
In the mean time I guess you could run one of your outbound head fittings into the bottom of the pipe. This just means that the water cooling the dry pipe jacket will already be warm but I guess it's no big deal there cuz the water stays pretty cool. Our water is at 75F right now so I need cool water in my pipe.
Drilling and tapping the pump is cake so don't hesitate. You will just need a 90 degree 3/8"NPT fitting, a 3/8" tap, and whatever drill bit it calls for. You will see the spot on the pump where it needs to go. It is a little elevated round spot directly opposite of your current 90 degree fitting.
At this point I would get a 1100GPH bilge with the 1 1/8" pisser and hose because you will no longer have a stock bilge.