Your Tranny guy is right. Running thru the rad can magnify problems if your tranny is slipping which creates heat it will transfer that heat to the motor and cause it to heat too. If you have a motor thats running hot you will transfer that heat to the trans . The coolers in the rad can develope leaks also which will screw up your fluid and tranny internals. Both Chevy and GMC finally swapped to aux coolers on there trucks becuse of this. Run a good size cooler all by it's self. I'm doing the big B&M stack plate and I'm upgrading to larger 3/8 tubes.
