An update on Heisenberg and Kac-Moody categorification

Heisenberg categories act on many Abelian categories appearing in type A representation theory. There is also a general procedure to construct from a Heisenberg action another action of a Kac-Moody 2-category for some associated Cartan matrix. One of the adjunctions on the Kac-Moody side is matched up in an easy way with adjunctions on the Heisenberg side, but the second adjunction is much harder to describe. In this paper, we derive explicit formulae for this difficult adjunction, leading to some further simplifications to the existing theory.