Converse theorems and the local Langlands correspondence in families

We prove a descent criterion for certain families of smooth representations of $$ (F a p-adic field) in terms of the $$ -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903-4936, 2016). We then use this descent criterion, together with a theory of $$ -factors for families of representations of the Weil group $$  (Helm and Moss in Deligne-Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth $$ -modules (the so-called "integral Bernstein center") in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural "local Langlands correspondence in families" of Emerton and Helm (Ann Sci Éc Norm Supér (4) 47(4):655-722, 2014).