Heater Core Hose to Flush Coolant? Left or right?

I'm in the process of flushing the coolant, but I'm not certain which hose is the correct one to attach the T fitting too. Am I right in saying it's the one on the left? That appears to be the one running straight into the block. Also, what is that "thing" that is inline on the tube? I took it off for a minute and stupid me wasn't paying attention to which direction it was. Is it in the correct direction now?

.. And if that is the correct hose, can I remove that "thingy" temporally, and but the T in it's place?

Thanks.

That is the coolant bypass valve--yes you van put a T in there--

The arrow should point to the firewall when you re-insatll it.

The only real thing it does, is block coolant flow to heater core, when AC is on. The truck will be fine without it. Plug the vacuum line though.

Excellent, saves cutting a hose.

Now when I put it back on, the image below I found on another thread (the PCV valve, Sea-Foam and Tune-Up 1999 Explorer 302 AW thread) shows the tiny hose pointing out, away from the firewall. But if I line the arrow so they face the firewall, it will be facing opposite to the direction in that picture.
http://www.explorerforum.com/data/3526/DSCN4676.JPG

I meant--the arrow should point in that direction of flow--from it to the heater core--

When I turn the engine on briefly, the water flows towards the firewall. So like you say, it would make sense that the arrow point towards the firewall. But then it would seem to be backwards from what that picture above has. And in fact, backwards from what is in the image near the front of the Hayes manual.

Wish I was looking when I removed it.

Can anyone confirm the direction... like I say, I would think the arrows should be facing away from the firewall... but then it would be facing the wrong direction from the manual and above picture. I'm assuming if I put it in the wrong way that would bad....

The hose coming off the thermostat housing

Here's another image of the valve,
Again, that appears to have the arrows facing away from the firewall.

And another picture


I don't understand why it seems the arrows go the oposite way from the direction of flow......