On Equivalences of Derived Exponential Functors

A strong symmetric monoidal functor $F\colon (Ab^{free,fg},\oplus)\to (Mod(k)^{flat},\otimes)$ is determined by the Hopf algebra $F(\mathbb{Z})$ over the ring $k$. We will show that the algebra structure on the left derived functor $L^* F(P)$ can be recovered from the augmented coalgebra structure on $F(\mathbb{Z})$ for $P\in D^{>2}_{perf}(Ab)$. Using a similar technique we will prove that the multiplicative Dold-Puppe-Thom isomorphism $H_*(K(A,n);\mathbb{Z})\simeq L_*Sym A[n]$ is functorial in $A\in Ab^{fg}$ whenever $n\ge 2$. By contrast, if $n<1$, this is known to be false in general.