Faucets don’t magically produce water of different temperatures on their own. They are valves that open and close pipes, the cold directly from the main source of water and the hot from a device that heats water.
The device that heats water (a “water heater”) has hot water within it that is maintained at increased temperature waiting on use. But the pipe leading from it to the faucet is full of unheated water which needs to be pushed out ahead of the hot water. This is why it takes time for the water from the faucet to get hot.
The cold water source is always unheated. Opening it always gives cool water instantly.
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