There are many instances where mapping a file texture to hair is useful. For example when rendering tiger or leopard fur. This requires texture coordinates from the surface mesh.
By default Arnold generates UV coordinates in which the V coordinate goes from 0 to 1 along the length (height) of each hair and, when using the ribbon mode, the U coordinate goes across the width of the hair. The following image is obtained by mapping a red to blue ramp texture.
As you can see in the side view of the hair below, the tiger pattern is defined along the UV of each individual hair fiber length as well as across the object's surface UV's.