Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy category?). Now, in the well-known announcement paper "Ordinary $RO(G)$-graded cohomology" (by G. Lewis, J.P. May, and J. McClure; https://projecteuclid.org/euclid.bams/1183548004) to any G-CW spectrum $Y\in SH_G$ a complex $C_*Y$ whose terms are Mackey functors (i.e., contravariant functors from the orbit category) was associated. Using this functor one easily defines an (equivariant) cohomology functor $H(-,M)$ with values in abelian groups corresponding to any Mackey functor $M$.
My question is: where can I find details on these functors? In particular, does the association $Y\mapsto C_*(Y)$ gives an exact functor from $SH_G$ into the derived category of Mackey functors?