Title: Coaction functors, II,
Abstract: In further study of the application of crossed-product functors to the
Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other
properties that crossed-product functors may have. Here we introduce and study
analogues of these properties for coaction functors, making sure that the
properties are preserved when the coaction functors are composed with the full
crossed product to make a crossed-product functor. The new properties for
coaction functors studied here are functoriality for generalized homomorphisms
and the correspondence property. We particularly study the connections with the
ideal property. The study of functoriality for generalized homomorphisms
requires a detailed development of the Fischer construction of maximalization
of coactions with regard to possibly degenerate homomorphisms into multiplier
algebras. We verify that all "KLQ" functors arising from large ideals of the
Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the
opposite extreme we give an example of a coaction functor having none of the
properties.