Help with cool tube setup

So I'm setting up for my second grow one of the main changes I want to make is improving my ventilation. I have the right supplies I think I need to arrange them better.

I'm running a 600w mh/hps in a 4x4x6.5' tent. I put the ventilation in without enough research the first time around. Luckily it was winter so I was able to keep my temps down. I didn't use a carbon filter so I ran an 6" booster fan (220 cfm) from the window pushing cold air over the cooltube out the other side of the tent. I also ran a 450 cfm inline fan for exhaust out of the top of the tent.

This time around I'm adding a filter and have been researching the best way to arrange everything. I will also be adding a window ac unit to cool the room.

I want to run ducting from the ac to the cool tube which will connect to the carbon filter then out the top exhausting with the inline. I've read it is better to run the carbon filter to the cool tube to the fan directly but I'm not clear why that's better as you'd be running the tent air through the tube. With the ac unit would it make it better to run the cold ac air straight to the cooltube with the filter after? Sorry if this is hard to understand but any help would be great.

Also I'm considering using the extra boost fan to bring cooled air from the ac during the hot months of the summer but I'm not sure if that will create much pressure. It's 220 cfm and the inline is 450 and pulling through a carbon filter.

any thoughts?

OK the carbon filter is to filter air to remove odors - period. That would be the air IN the tent. The air over your cooltube should not be that air. In fact that air should have no odor. Fan in front of the light fixture to push air through rather than pulling heated air and burning fans and bearings. As to cooling your fixture with your AC - that light is obviously the source of most heat issues so handling that in this manner seems okay.

I know I shouldn't be running hot air though the tube but what exactly is the difference between pulling the air from the ac through the tube and pushing the same ac air through the tube?

Pulling means the fan is pulling now-heated air through it. Heat and vibration are the 2 main culprits in mechanical failures. Pushing cooled air over the tube and out is the goal here. Saves fan motor bearings and shafts.

Okay I get it the point is to not damage the fan. So I assume this means I will need a seperate fan for exhaust that will pull through the filter and I can't combine my main exhaust and cool tube on the same track correct?