Trainer following the horses? (5)

I believe the answer is:

coach

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'trainer' is the definition.
(coach is a kind of trainer)

I don't understand how the remainder of the clue works.

(Other definitions for coach that I've seen before include "Type of bus" , "Bus, instructor" , "Teacher of sporting activity" , "Trainer and train carriage" , "Single-decker bus" .)

I've seen this clue in the Evening Standard.

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