Is this correct?

Fuel is pumped from the tank via the lift pump to the filter and then to the tandem pump, to the injectors. Unused fuel then goes back to tandem pump to be returned through fuel temp sensor to filter and then from filter to fuel cooler and then back to the tank. Yes? Thank you for your input.

Yes,
Once the fuel warms up to the required temp, the thermal tee on the filter routes the excess fuel to the tank. Prior to that, the fuel returns to the filter and the thermal tee circulates the fuel back to the tandem pump to aid the fuel in warming up and keeping the fuel filter from clogging from gelling fuel in cold weather.

Thank you.