Explicit description of the right adjoint

Question

Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.

Given the functor $F$, we can also construct $F^{\prime}: C \to Cat_\infty$, $c \mapsto Mod_{F(c)}$, where $Mod_{F(c)}$ is the infinity category of modules over the ring $F(c)$. Then we have an adjunction $$Mod_R\rightleftharpoons\lim_C(c \mapsto Mod_{R(c)}).$$ The map from $Mod_R$ is simply given by ``tensoring up'' through the diagram. For the right adjoint,I have heard that it can be written explicitly as a homotopy limit of a diagram of modules. I was wondering if anyone had any reference for this. To explain what I mean by explicit description I have a small example below.

Example: For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by $$L_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i_K ] {}}} K$$ where $g$ is a topological generator of $\mathbb{Z}_p^{\times}$ and $\Psi^g$ denotes the Adams operation. This diagram gives rise to the adjunction $$L_{K(1)}Sp \rightleftharpoons \lim (Mod_K \rightrightarrows Mod_K)$$ In this case the objects of the limit category are given pairs $(M,\phi)$ where $\phi : M \wedge _{\psi^g} K \to M$ is an isomorphism. The right adjoint is simply given by $lim(M \xrightarrow{\phi \circ \psi^g -1} M)$ ie the equaliser of $\phi \circ \psi^g$ and $id$. I am looking for a similar description for bigger diagrams with more simplices. Thank you in advance.

Answer 1

See Theorem B in On conjugates and adjoint descent, Horev-Yanovski, which states exactly this. The statement doesn't include this, but the body of the paper describes explicitly in what way the right adjoint is a limit of the right adjoints.

Note that your example is not a special case of your general setup because $Mod_{L_{K(1)}S}$ is not equivalent to $L_{K(1)}Sp$.

Answer 2

See (large parts of)

Mathew, Akhil 
The Galois group of a stable homotopy theory.
Adv. Math. 291 (2016), 403–541 ,

(arXiv link) where the case $n=1$ of Proposition 10.10 is close to your example.