I will present some results showing that the Hubbard repulsion and other interactions like the Hund's coupling and electron-phonon interaction can either compete or cooperate, leading to different strongly correlated phases.
First I will review the interplay between Hubbard U and Hund's coupling emphasizing how different fillings are characterized by different scenarios. Namely at half-filling the two interactions cooperate to localize the electrons in a high-spin Mott state, while for other integer fillings they compete, leading to a Hund's metal region bridging two strongly correlated insulators .
I will then address how an electron-phonon coupling changes the picture. In a two-orbital model we witness a cooperation between the Hubbard U and Jahn-Teller phonons, while the latter compete with the Hund's coupling. Also in this case we deal with the competition between two Mott insulators bridged by a hybrid insulator with mixed properties 
A. Isidori, M Berovic, L. Fanfarillo, L. de' Medici, M. Fabrizio and M. Capone, Phys. Rev. Lett. 122, 186401 (2019)
 A. Scazzola, A. Amaricci and M. Capone, arXiv:2111.14663