They didn't. There is no such thing as a family coat of arms - this is a modern scam developed by fake genealogy tradespeople who prey on ignorant and gullible customers. Just because your surname is Fitzwalter or Grosvenor (for example) you would have absolutely no claim to the Fitzwalter or Grosvenor coat of arms unless you are heir to the title and estate (and you would certainly be aware of that).
Heraldry has always been issued on an individual basis, mainly to knights when they win their spurs or are knighted for outstanding deeds. A knight would pass on that heraldry to his eldest son, but nobody else in his family (and nobodywith the same family name) is entitled to use that heraldry.
So a man might have a specific coat of arms and his brother might have an entirely different coat of arms. His wife, who might have entitlement to her own coat of arms, might combine her own and her husband's heraldry in a lozenge (not a shield).
Modern English knighthoods are generally awarded for outstanding service in business, sport or politics; these are generally not hereditary, so the new knight would have a coat of arms along with his title, but he can not then pass it on to his eldest son.
Coats of Arms belong to individuals, not to families.
The coat of arms can belong to various entities, including individuals, families, cities, or nations. Each coat of arms is unique and represents the history, achievements, and values of the entity it belongs to. For example, royal families often have their own coats of arms that signify their lineage and authority. If you have a specific coat of arms in mind, please provide more details for a more precise answer.
Coat of arms.
Names do not have crests or coats of arms. Some individuals have coats of arms. Some families have crests; and some families of the same name do not.
Coats of arms are assigned (held by) individuals not surnames or whole families.
it was used to protect theirselves in battle or show pride in their families.
Coats of Arms are associated with particular people or families of prominence, such as royalty and nobles, not with a surname.
There is no widely recognized coat of arms specifically associated with the surname "Cayot." Coat of arms are typically tied to specific families or regions, and while some surnames may have heraldic symbols, not all do. If you're looking for a coat of arms related to a specific lineage or region, it may require genealogical research or consultation with heraldry experts.
Coats of arms identify individuals, not families. As they are known today they were developed in medieval Europe. The Asian cultures developed other means of identifying nobles and other leaders on the battlefield.
Unfortunately, many families have chosen not to show a motto on their coat of arms.
The College of Arms is the official repository of the Coats of Arms of England, Wales, Northern Ireland and the Commonwealth. Other nations have similar organizations. Coats of Arms are authorized by senior Heralds on letters patent. A Coat of Arms with royal connection may carry elements of the royal coat of arms within it. There are several websites which list coats of arms, families and their connections; DeBretts being a good example.
The plural of "coat of arms" is "coats of arms".