  We say that a fusion system is the composition product of two subsystems if
every morphism can be factored as a morphism in one fusion system followed by a
morphism in the other. We establish a relationship between the characteristic
idempotent of a saturated fusion system that is the composition product of
saturated subsystems and the characteristic idempotents of the component
systems. Consequently we obtain a compatibility result for transfer through the
composition product and transfer through the component systems.
