The rooster's name is Kureno, but we never see him transformed, for reasons explained in the manga.
The horse's name is Isuzu, or Rin, as she is better known by.
Neither of these characters are in the anime--they are only present in the manga series.
The rooster in the Chinese zodiac is Ayame Sohma in Fruits Basket. He is portrayed as confident, flamboyant, and passionate about fashion and design. Ayame is one of the members of the Sohma family cursed by the animals of the zodiac.
Kureno Sohma
The rooster in the zodiac in "Fruits Basket" is Ayame Sohma. He is the younger brother of Yuki Sohma and represents the zodiac animal for the Year of the Rooster. Ayame is charismatic, flamboyant, and often provides comic relief in the series.
Isuzu "Rin" Sohma is the horse from the Chinese Zodiac .
Kureno is a character in the manga "Fruits Basket" who is part of the Sohma family cursed with the Chinese zodiac spirits. He is represented by the rooster. Kureno plays a significant role in the story as he is mysterious and has a unique connection with one of the main characters.
Rin is Haru's girlfriend. She is a member of the Chinese Zodiac, she represents the horse
The Rooster
Ritsu Sohma from Fruits Basket is the monkey of Zodiac.
sheep rooster
1969 was the Year of the Rooster.
The rooster is a zodiac sign in Chinese astrology. Some traits associated with the rooster are observant, sociable, outgoing and attentive to detail.
Kyo Sohma
He is the cow/ox.
In "Fruits Basket," Kagura is the boar of the zodiac.