Second adjointness and cuspidal supports at the categorical level

We prove the second adjointness in the setting of the categorical local Langlands correspondence. Moreover, we study the relation between Eisenstein series and cuspidal supports and present a conjectural characterization of irreducible smooth representations with supercuspidal $L$-parameters regarding geometric constant terms. The main technical ingredient is an induction principle for geometric Eisenstein series which allows us to reduce to the situations already treated in the literature.