The air isn’t cold.
The movement of it across a wet surface (e.g. your skin, your sweat, etc.) causes water to evaporate and moves the evaporated water away. Cooling a human is about facilitating evaporation – the human sweats, air moves across, which allows the water to evaporate, which cools the body (relatively speaking). Humans sweat, dogs pant, and moving air makes the process more efficient.
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