I'm pretty sure in the opening chat with the professor he explains that you fight with the hentaimon's power. So you (as of now) won't be summoning the hentaimon at all. The equip function was kinda confusing to me as well, but you equip them from the menu and then you can use whatever ability that specific character has (each evolution gives you access to a new move). I think most of this is actually explained via the dialogue it puts you through.