A cold water wash will not only remove any residual lactose on the curd surface but will slightly rehydrate the curds with water and dilute (thus squeeze out) more of the whey inside.
Note: Curds washed in cold water will cause the water to flow into the curd and is just the opposite of adding warmer water to curds. This causes the moisture to flow out and thus drying the curd as in the process of making Gouda cheese.
By the time the curds have drained, the temperature has likely dropped from the cooking temperature down to the mid 90°F range.
For washing you want to use a Non-Chlorinated water that is not too alkaline, so if you have a very soft water it would be best to use a bottled water for the washing.
The water should be as cold as possible because that will help bring it down to the fridge temperature much faster.
The goal is to wash and chill the curds as quickly as possible.
The cold water should be changed several time to dilute the lactose that is being washed off the curds as well as keep it from being warmed up too much by the warmer curds.
As shown above I find it easier to wash the curds by submersing the colander with curds into the cold water.
I find my tap water here cool enough for this but commercially they use refrigeration cooled water in a downward cascading and ending with the final water at just a few degrees above freezing.
Drain all of the curds in the colander when they have been well washed and cooled sufficiently.