750E Transmission question

It's really a simple question. Is there a thermostat?
My 2006 2WD V8 SR5 DC has a transmission cooler. That is, it has the little mini radiator at the front.
I want to do a drain and fill on the transmission. Most videos I see say to 'pin the thermostat if you have a transmission cooler', but I cannot find one on my transmission. The only real 'attachment' I can see on the passenger side that is anywhere near the coolant pipes is the 'neutral start switch assembly'.

Given the age of these trucks, is it possible that it just doesn't have a thermostat and the lines to the cooler are just open all the time? I just want to make sure to do the job properly!

ETA = When I say Coolant lines, I misspoke. I meant lines HEADED towards coolant from the transmission casing.

 

No t stat

 

Thank you!!!

 

Well, that makes it a pretty simple exercise then. The fill plug on the 750E is much more accessible and even a 'comfortably built' person like me can easily get to the three plugs I need to get to with the wheels on the ground.

 

Just gonna tack this on to this thread, because i’m too lazy to check, and dont really need to know right now, but does the fluid hit the cooler before of after the radiator?

 

check the FSM downloadable; see #4 here:
https://www.tundras.com/threads/valve-cover-bolt-torque.120114/#post-3082070