Good Transmission Pumpout

Hey Folks,
I'd like to see a diagram of the transmission/cooler flowpath for the 2019 gen 3 v6 Taco as when I do a full replacememt with filter I can catch all the old stuff and eliminate as much of it as possible. One of my questions is, when I pin the thermostat open, does that effectively create another individual loop to the aux cooler, or does it all flow through the whole system, radiator and aux as one loop.
Another question is, what is the big can-shaped looking thing on the transmission passenger side? is it a transmission to engine coolant heat exchanger?
I've done the flush on a couple of gen 3s but I never feel I got ALL of the old fluid out.
If anyone has traced this or don it please let me know.

Thanks so much,
Jim