The hydramatics I worked on back in the 1950's had a small heat exchanger mounted beside the transmission, with short hydraulic lines from the transmission to the cooler unit, and heater-hose sized tubing plumbed from the engine to the cooler. I don't think I've ever seen one with coolant going into the transmission itself. It's possible there might be a loop of tubing going through the pan where heat transfers to the coolant, but it's probably a pretty scarce setup if that's how yours works.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!