The structure you're referring to is likely the ureter. However, it is important to clarify that the ureter is not part of the large intestine; rather, it is a tube that carries urine from the kidneys to the bladder.
The large intestine itself is made up of several parts, including the cecum, colon, rectum, and anus. The cecum, located at the beginning of the large intestine, does have a small tube called the appendix attached to it, but this is not involved in the process of turning waste into urine.
If you're specifically looking for a part of the digestive system that plays a role in the formation of waste, you should look at the kidneys for urine production and the ureters for urine transport. The large intestine's role is primarily in absorbing water and electrolytes from undigested food.