Relating dgLa and cdga: $C^*$ and its adjoint. Part 2

In the previous talk we studied the existence of a right adjoint $D$ to the cohomological Chevalley-Eilenberg  functor $C^*$ between the $\infty$-categories $dgLa[W^{-1}]$ and $(cdga^{aug}[W^{-1}])^{op}$. We also discussed that $C^*$ applied to free dgLa's is (homotopically) very simple: It is equivalent to applying the functor that sends a cochain complex $V$ to the augmented cdga $k\oplus V^*[-1]$, equipped with the product determined by $(1,a)(1,b)=(1,a+b)$.

In this talk we will use this fact to get an explicit description (up to homotopy) of the functor $\#\circ D$, where $\#$ is the forgetful functor from the infinity category $dgLa[W^{-1}]$ to the infinity category of cochain complexes $dg[W^{-1}]$.

We will also see that for sufficiently nice dgLa's $\mathbf{g}$, we have an equivalence $D\circ C^* (\mathbf{g}) \to \mathbf{g}$.