The process involves extension and invagination of the plasma membrane to form small vesicles in pinocytosis (pinocytotic vesicles) or vacuoles in phagocytosis (food vacuoles).
Note the clear vacuoles in the colloid next to the epithelium where the increased activity of the epithelium to produce increased thyroid hormone has led to scalloping out of the colloid.
Finally, plant cells also contain structures known as vacuoles; these are filled with a watery substance which contains dissolved sugars called cell sap.
Their relationship reminds us a bit of Paramecium bursaria and the green algae Chlorella that live as endosymbionts inside of vacuoles specially designed to hold them.
Among heterotrophic single-celled eukaryotes, food is taken into the cell in various methods but once it's there, it's wrapped by a membrane and forms something called a food vacuole.
Then, Ohsumi altered different genes, to figure out which ones were responsible for sending the old parts to the vacuole in the first place -- this is the actual autophagy process.