The shogun is the commander-in-chief of the Emperor’s army. In practice, at any given time, this is only a small Imperial Guard in constant service to the Emperor. But in theory, the Emperor can raise levies from his vassals and create a massive army—-which the Shogunate would control.
The title of Shogun, like the position of Imperial consort, is a prize to be fought over by the seven clans. The Shogun must have a good relation with the Emperor, or else their position of power as the Emperor’s right hand would be diminished by infighting. But the Shogun has a free hand to meddle with the affairs of the clans. Thus holding the Shogunate is a great coup for the ruling family and their clan.