I don't know what article are you referring to, but here is the anatomy story of the liver/colon links:
Our colon have the ability to absorb water with water soluble matter. Water absorbed is carried from the colon to the liver by portal circulation blood, by portal vein (blood vessel).
Almost all of the blood coming from the digestive system drains into a special venous circulation called the portal circulation. This is because it contains all the nutrients and toxins that have been absorbed along the digestive tract from ingested food. Before these absorbed substances can go into the systemic circulation (the main blood circulation in the body), it must be filtered first to remove or "detoxify" toxins first. This filtering and detoxification is one of the functions of the liver.
So, any liquid (not just coffee) introduced into your colon by enema, if absorbed, will be carried directly into your liver in almost no time.
Now, once coffee reaches your liver, it stimulates production and excretion of bile, including excretion of bile from your gallbladder.
Coffee is not the only herb that stimulates production and excretion of bile. There are virtually hundreds of thousands of herbs that have that ability. All those herbs are called cholagogue.
Here are few common cholagogue herbs:
, sage, golden seal, lobelia, Swedish bitters, Chinese bitters, copits, gold coin grass, burdock root, and all herbal oils like olive oil, walnut oil, ....
Now, let's go back to your liver.
Once coffee reaches your liver, your liver excretes bile into your duodenum (through common bile duct).
That bile goes from your duodenum through your whole small intestine, and through your colon, and some of it, together with enema, leaves your body (depends on how long time you do enema, how many times you repeat water infusion)
So, "the Duct between the sigmoid colon and liver" is called Portal Vein.
"the Duct between the liver and sigmoid colon" is your billiary tree and your intestines:
Liver -> Intrahepatic bile ducts -> common bile duct -> duodenum -> rest of the small intestine -> asc. colon -> transverse colon -> desc. colon -> sigmoid colon -> anus -> back to nature.