Ideally, cool the water in the container used for boiling. But boiling for 10 minutes is often impractical, especially in the wilderness where time and fuel are considerations. Other fuels are difficult to transport in the wilderness. People in the wilderness generally use two liters of water a day. Most pathogens are killed at lower temperatures.
Thermal death is a function of both time and temperature.
Lower temperature is effective with longer contact time. Water can be considered pasteurized when it comes out of the tap too hot to touch or when it is merely brought to a momentary boil. The necessary contact time takes place in the water heater and pipes. The contact time for momentary boiled water occurs while the water is brought to the boiling point plus the cooling time. Pasteurized water is not sterile.