Bi-directional association between Footballer Mapper and Bowler Mapper

This question is about a specific UML diagram in the book called Patterns Of Enterprise Application Architecture, by Martin Fowler.

Why is there a 'bi-directional association' between Footballer Mapper and Bowler Mapper in the 'Inheritance Mapper' Class Diagram on page 302?

There isn't.

You're looking at an aggregation from the Player Mapper to one of the concrete mappers: Footballer Mapper, Bowler Mapper and Cricketer Mapper.

If you read the 2nd paragraph on page 304, he explains this design choice.