On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper *Algebraic K-theory and étale cohomology*, which reads

$$H^p_{\acute{e}t}(X, \mathbb{Z}_l(-q/2)) \Rightarrow \pi_{-q-p}\hat{L}KX$$

where $KX$ is the algebraic K-theory spectrum of $X$ and $\hat{L}$ denotes the $\ell$-completed Bousefield localization at topological K-theory, implies the natural isomorphism

$$\pi_i\widehat{L}(KR) \cong \pi_i\text{Map}_{\Gamma_F'}(X_+^\theta, \hat{\mathcal{K}})$$

where here, $R$ is the ring of integers localized away from $\ell$ of some totally real field $F$, $X$ is a space realizing the étale homotopy type of $R$, $\theta$ is the character of $\pi_1(X)$ corresponding to the $\ell$-adic cyclotomic character, $X^\theta$ is the cover of $X$ corresponding to the kernel of $\theta$, with the corresponding action of $\theta$ of the Galois/fundamental group. The subscript plus, as usual, denotes the unreduced suspension spectrum. The Galois group of the $\ell$-adic cyclotomic extension is in particular $\Gamma_F'$, and lives inside $\Gamma'\cong \mathbb{Z}_l^\times$ (i.e. the corresponding Galois group over $\mathbb{Q}$.)

The $\text{Map}$ is then an equivariant mapping spectrum, with the action of $\Gamma'$ via Adams operations on the target.

(It is assumed we have a rigid, i.e. not just up to homotopy, action of $\Gamma'$ on $\hat{\mathcal{K}}$.)

So, **how does this follow**? There is no real explanation given, which makes me think it has to be simple. But I'm confused what the relationship of the localized algebraic K-theory groups to étale cohomology has to do with this twisted equivariant mapping spectrum.

Apologies if this is some very formal thing; I'm pretty un-versed in these matters. I'll delete this question if it turns out to just be very obvious somehow.