foal by (foal’s sire) out of (foal’s dam) by (foal’s dam’s sire)
Depending on who you are talking to, it will then go (foal’s dam’s dam’s sire) then (foal’s dam’s dam’s dam’s sire) possibly followed by the famous female family name or number. -If you are talking to European WB breeders.
The reason being that hopefully whoever is listening knows what those sires get in their offspring, so as to predict how the foal may turn out. And showing the consistent breeding to top sires over generations of the tail female/bottom line.
Other breeders will pull out a copy of the pedigree and note the top producing dam’s on sire’s side and dam’s side of the foal, or a breed building sire of famous horses. They are again noting where talent shows in the pedigree and hopefully the foal.
Non-European WB breeders do tend to name the individual dams on the bottom, since a sire may have many daughters and there can be much difference in quality even between full sisters.