"Yun Bei" is what we call in Taiwan to this pitcher to serve tea. Yun means "to make even" and "bei" means cup or bowl, so in English it could be called a "recipient to make even" the tea.
 
Its purpose is that all guests drink the same tea, since by serving it from the teapot, the Yun Bei will balance the flavors and aromas of the top of the teapot with those of the bottom. You can do the test of serving directly from the teapot to two glasses, one after the other, and you will notice differences in the tea of each of them. Hence the reason to use this accessory when preparing in Gongfu brewing style.
 
This Yun Bei holds 380 cc and is perfect for large kettles 150 cc or larger, you can find our teapots and Gaiwans of that size in this link.