Hi,
This model year, there isn't a cold start solenoid. Toyota used them years ago, but I haven't seen one on anything this new. I could be wrong, but there is nothing listed in the service manual indicating one as well.
The coolant temperature sensor is what signals the computer temperature. With that information, the computer determines how rich or lean to make the fuel mixture for an easy start and then running. Has the check engine light turned and stayed on while the engine is running?
What should be done, and it's simple, is to use a live data scan tool to check the temperature signal from the sensor before starting the vehicle. It should be very close to ambient temperatures.
I attached a pic below showing the location of the sensor. It may be a good idea to disconnect the electrical connector and make sure there is no corrosion or damage to it.
Let me know what I can do to help or if you have other questions.
Take care,
Joe
See pic below.
Image (Click to make bigger)
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Wednesday, October 5th, 2022 AT 8:25 PM