A takye, takye, or tekke was a skullcap worn under a headdress to protect the headdress from sweat and grease. Poor men sometimes wore skullcaps as their only headgear.
In the picture, the diner on the right has removed his turban (on the floor beside him, under a turban cover) and is wearing only a skullcap.
This drawing was done by a European who may not have been to Turkey, so it may not reflect actual practice. Turkish-drawn art never shows men as anything less than fully dressed at the table. (It also shows men with their turbans off and nothing underneath, so the skullcap may not have been universal.)
