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If your C6 has an automatic transmission, the outlet from the transmission is the fitting that attaches to the upper part of the radiator. The lower radiator fitting attaches to tubing that goes back to the transmission.
If your going to eliminate the trans cooler in the car's radiator and install a stand-alone transmission cooler, plumb the lines into the aftermarket cooler either way (don't make any difference)
I know nothing about a separate engine oil cooler ... I know my GS don't have a cooler for the engine oil.
Oil flows from the oil pump at the front crank drive, through the block to the front hole in the pan over the filter. Then through the filter and back up to the front hole in the side of the pan. At that point the flow is either out to a cooler or makes a U-turn and goes back in the rear side hole of the pan and then back up to the block and throughout the engine again. If oil went out to a cooler, it returns to the rear side hole in the pan.
This shows the direction of the flow, but fails to show where the cooler lines would attach or the U-turn bypass is mounted.
If you buy an aftermarket cooler package from someone like Improved Racing, it'd include the casting to replace the U-turn bypass. Otherwise get it from GM.
Not when it comes to which way you plumb the fluid into the cooler. The only concern is to avoid a trapped air bubble.
Yep, supply to the bottom/lower fitting of the cooler, and return out the top/upper fitting of the cooler fittings, then way any air is pushed out the top of the cooler and back to the trans/motor on the return lines. If you pipe the supply into the the top of the cooler, and return out the bottom, you will trap air in the cooler.