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He debuted in Plaza México on July 12, 1966, with Gonzalo Iturbe and Leonardo Manzanos. He took his alternativa in Plaza Monterrey, Nuevo León, México, on August 28, 1966. His Padrino was Antonio Velázquez. His Testigo was Manolo Martinez. His first bull was named "Cariñoso", from the ranch of San Miguel Mimiahuapam. He took his confirmación in Plaza México, on January 14, 1968. His Padrino was Alfredo Leal. His Testigo was Jaime Rangel. Bulls were from the ranch of Jesús Cabrera. He took his confirmación in Madrid, Spain, on May 22, 1971. His Padrino was Miguel Mateo "Miguelín". His Testigo was Gabriel de la Casa. Bulls were from the ranch of José Luis Osborne.
Unlike some matadors who are well known for their hard partying lifestyle, Eloy Cavazos is known and respected for his professionalism and a disciplined lifestyle that he has maintained throughout his career. This pattern was evident from his beginnings with his late mentor, the Matador Jaime Bravo, who would make sure that Eloy did not drink, went to bed early, and woke up early to train.
Eloy Cavazos and Jaime Bravo were in a tragic 1970 automobile accident, near Zacatecas, México. Eloy survived. Jaime died, shortly thereafter. Their driver died in the accident. The accident occurred after a bullfight, in Zacatecas, where Jaime Bravo had just won that year's "Golden Sword" award. The sword disappeared in the accident. Eloy Cavazos's manager, Rafael Baez, had also been the late Matador Jaime Bravo's manager.
The Mexican Matador Alejandro Amaya is regarded as Eloy Cavazos's protégé.