My above experience at the new Siler City Supercharger brings up the question, What exactly is the car doing to "prepare the battery for supercharging" as it navigates to the supercharger? The manual says, "Model 3 pre-heats the Battery to ensure when you arrive at the Supercharger, the battery temperature is optimal and ready to charge. This reduces the amount of time it takes to charge." But it doesn't say what that optimal temperature is, although some older information from Tesla says their battery management system normally tries to keep the battery below 95°F (35°C) and tries to keep its lifetime average at 77°F (25°C).
However, the situation is different when DC Fast Charging the battery, since the battery does charge faster at higher temperatures. There's another TMC forum discussion here called "How hot should the batteries get while charging?" with a comment from Zoomit that says, "The target temperature for On Route Battery Warmup is 40°C (104°F), based on a comment by a Tesla engineer at the V3 Supercharger test station when it first opened in Fremont. I think the battery will charge at max rate when that temp has been reached..."
For that long discussion see:
How hot should the batteries get while charging?
But once the car reaches the above "target temperature" of 40°C to allow faster charging and is hooked up to the Supercharger, the car's battery management system will try to keep the battery temperature below 45°C (113°F) as the DC Fast Charging
by itself heats up the battery. Once the battery temperature gets to 45°C during the DC Fast Charging, the Supercharger will then begin to reduce charging power significantly. For more on that see:
Watch Tesla's battery thermal management in action while Supercharging
To summarize, for fastest charging at a Supercharger, the car's battery temperature should be close to 104°F (which the car will try to take care of automatically if you navigate there), but if it's a hot day and the battery temperature gets up to 113°F, your Supercharger will reduce your charging power at that point. If it's a cool/cold day and you don't navigate to the Supercharger, which means you could arrive with a battery temperature much below 104°F, you will get reduced charging power.