They both have some relationship with the main characters.
Like the units MARs, Happy Rain♪, SoLaMi♡SMILE, TRiANGLE and MY☆DREAM, WITH is a unit consisting of a lovely type idol, a pop type idol and a cool type idol. Like the above-mentioned units, WITH has a lovely type idol as the center.
According to a special interview, Koyoi is the leader of WITH. He started it with Asahi at first, and then Shougo got recruited later on.
In the drama CD track from WITH album, Asahi said that Koyoi used to visit Yamipri frequently in the past. On WITH stage, it was revealed that Koyoi left WITH during that time and joined Shinya's Yamipri unit, DARK NIGHTMARE.