On Equivalences of Derived Exponential Functors

A strong symmetric monoidal functor $F\text{\hspace{0.17em}⁣}:(A{b}^{free,fg},\oplus )\to (Mod(k{)}^{flat},\otimes )$ 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 $L^{\ast }F(P)$ can be recovered from the augmented coalgebra structure on $F(\mathbb{Z})$ for $P\in D_{perf}^{>2}(Ab)$. Using a similar technique we will prove that the multiplicative Dold-Puppe-Thom isomorphism $H_{\ast }(K(A,n);\mathbb{Z})\simeq L_{\ast }SymA[n]$ is functorial in $A\in A{b}^{fg}$ whenever $n\ge 2$. By contrast, if $n<1$, this is known to be false in general.