Set up the Unsorted/adjoint (categorical) \begin{align*} \mathsf{D} \coloneqq\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \coloneqq\mathsf{C} .\end{align*} Then \(LR \coloneqq p^*p_*\), and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{ { \, \xrightarrow{\sim}\, }} {\mathsf{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}

Theorem: limit of \(F\) in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\) is a homotopy limit of its Unsorted/adjoint (categorical) in \({\mathsf{sSet}}{\hbox{-}}\mathsf{Cat}_{{\mathsf{Kan}}}\), and the limit of its adjoint in ${\mathsf{sSet}}^{\Delta^{\operatorname{op}}}_{ \mathsf{CSS} } $.

Finding Unsorted/adjoint (categorical) is usually easy, because checking isomorphisms on hom sets is concrete.