I believe that the best way to assemble and fix a movement is to first understand how everything about it works. So I made some animations to help in understanding how the chronograph assembly works.
The Seiko 6139 features a column wheel and a vertical clutch. The chronograph second engages and engages with the 4th wheel via a clutch spring. The clutch is controlled by the first and second coupling levers.
See how pressing the operating lever rotates the column wheel, which in turn, controls the coupling levers.
While the chronograph is running, pressing the flyback lever will have no effect.
But when the chrono is stopped, pressing the flyback lever will activate the hammer, which in turn, resets the chronograph second wheel and the minute recorder wheel.
I hope these animations were helpful.