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Color temperature is the appearance of a black body at that temperature. For example, 2400K is a tungsten bulb, 5500K the sun, etc. In Adobe Camera Raw or Lightroom, the temperature slider is making the picture more blue if you reduce the color temperature, rather than more yellow/orange.

Why is this so?

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White balancing when implemented correctly (like in Lightroom) ensures that whites stay white under the new illuminant, essentially mimicking the chromatic adaption performed by the human visual system. When you white balance a photo in Lightroom you are effectively picking an illuminant / whitepoint colour temperature for your photo. The result of that illuminant change is that colours get shifted toward blueish when lowering the colour temperature and they get shifted toward reddish when raising it.

As an example, assuming that you shot something with a Candle / Tungsten light source (2700K-3000K) and your camera white balance was set to Daylight (5500K-6500K). In order to neutralise the huge orange cast in your image you would have to introduce a lot of blue, you would do that by setting the colour temperature slider to the colour temperature of your Candle / Tungsten light source (2700K-3000K).

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Lightroom likes to use reverse psychology :) It does so in numerous places but it color temperature is one of the more obvious one.

You are right that diminishing the color temperature of an image should make it more yellow. However, what Lightroom is doing is letting you specify the color of incident light to indicate how much correction is needed. As you slide the color-temperature down, you make the light cooler and Lightroom is adjusting the image to compensate by making it warmer. When you find the right settings an image which started as too cool should end up neutral.

The crop tool as the same issue which makes is counter-intuitive. If you move the drop area as with most image manipulation software, it simply follows. With Lightroom, moving the crop area left move the image contents right. It is just the converse perspective, moving a cropping region instead of the image within a cropping area, respectively.

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The slider in Lightroom is not used to adjust the color of the light that lit the scene. That light has already been recorded. Rather, it is setting the correction for the light in the scene. The correction provided by any raw converter needs to be the "reciprocal" of the light that illuminated the scene. That is, the correction needs to be on the opposite side of the color wheel at the same distance from the white center as the original color of the light that illuminated the scene.

So when you set the color temperature to 2500K you're not telling Lightroom to "add 2500K light" to the picture. You're telling it to "correct for the 2500K light" that was used to illuminate the scene that was photographed.

If the scene was lit by a very orange light then the correction needs to be blue to make the scene look neutral. If you add more orange to a scene that was shot under orange light you're just going to make it more orange.

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