The haw is the cat’s third eyelid a.k.a. nictitating membrane. You can see it in the corner of the eye near the nose. When it plays its role it moves sideways across the eye to protect it from damage and to lubricate the corneal surface by spreading tears across the cornea (the transparent front of the eye). The haw then returns to its resting position.
When cats are asleep with their eyes half open which happens sometimes the haw moves across the eye. Humans cannot move their third eyelid. It is a vestigial pink lump in the corner of each eye (the end near the nose).
In a healthy cat the haw is inconspicuous. However in ill and/or undernourished cats it because very conspicuous as it is permanently drawn across part of the eye. This must be a diagnostic symptom for a veterinarian as it tells her that the cat may be going down with a major illness. It should be of assistance to cat owners who are deciding whether or not to take their cat to the vet; I’d do it.
The reason why the haw is drawn across eye like this is because ‘there are shock absorber pads of fat behind the eyeballs which start to shrink if the animal’s health is below par’1.
The shrinkage causes the eyeballs to sink into the head and in turn the haws move forward to half cover the corneal surfaces.
When a cat has pink eye (conjuctivitis) an inflamed and red haw is visible. In the photo above the inflammation has subsided but the haw is still half drawn across the eye.
There are many photographs on the internet of very visible haws. They are rarely referred to by this word. Most authors refer to them as the third eyelids or nictitating membranes, an equally strange word.
Reference 1: Dr Desmond Morris.