Let $G$ be a simplicial group and consider the adjunction $$ Fr:sSets\rightleftarrows G-sSets:U $$ where the left adjoint $Fr$ maps a simplicial set $X$ to the free $G$-simplicial set $G\times X$ and the right adjoint $U$ just forgets the action.

An adjunction defines a comonad $T=Fr\circ U$ and hence a simplicial object in $G$-simplicial sets (i.e. a bi-simplicial set with a $G$-action) $\mathcal{T}(X):\Delta^{op}\to G-sSet$ with $\mathcal{T}(X)_n=T^{n+1}(X)$, if I understand correctly.

I have the impression that $\mathcal{T}(pt)$ is just the $\mathcal{E}G$-construction (see this other question, $\mathcal{E}G$ is a bi-simplicial space with a $G$-action whose homotopy colimit is the usual $EG$) $$ \ldots \begin{array}{c}\to\\\to\\\to\\\to\end{array}G\times G\times G\begin{array}{c}\xrightarrow{(\sigma,id)}\\\xrightarrow{(id,\sigma)}\\\xrightarrow{pr_{0,1}}\end{array}G\times G \begin{array}{c}\xrightarrow{\sigma}\\\xrightarrow{pr_0}\end{array}G $$ where $\sigma:G\times G\to G$ is the multiplication.

My first question is: Is it really true that $\mathcal{T}(pt)\sim \mathcal{E}G$?

My second question is what happens when some arbitrary $G$-simplicial set $X$ is plugged into $\mathcal{T}$ and my guess is, that $\mathcal{T}(X)$ is just $\mathcal{E}G\times X$ with the diagonal $G$-action where $X$ is constant in the new simplicial direction. Is this true? If not, what else is $\mathcal{T}(X)$?



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