Final answer:
The exclusive bargaining agent's legal duty after a representation election is to fairly represent all employees in the bargaining unit, in accordance with the Wagner Act.
Step-by-step explanation:
The legal duty of the exclusive bargaining agent after winning a representation election is to fairly represent all employees in the bargaining unit. In the typical context of labor law, particularly referencing the Wagner Act, this duty is known as the duty of fair representation. This duty necessitates that the exclusive bargaining agent, often a union, represents the interests of all the employees in the bargaining unit without discrimination and with good faith and diligence. The Wagner Act, as part of labor legislation, helped to define the relationship between unions, employers, and employees in the United States and was instrumental in protecting the rights of workers to organize and engage in collective bargaining.