In his paper Cohomology $C_\infty$-algebra and rational homotopy type, Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a $C_\infty$-algebra, and how this structure determines the rational homotopy type of the space. (The result has been mentioned before on MathOverflow, eg in these answers.)

I'm trying to follow the proofs, which are somewhat light on details. In particular, they rely on an adjoint pair of functors $$\Gamma\colon CDGAlg \rightleftarrows DGLieCoalg \colon \mathcal{A}$$ introduced in section 4.3, between the categories of commutative differential graded algebras, and differential graded Lie coalgebras. The functor $\Gamma$ is given as the composition $$\Gamma\colon CDGAlg\stackrel{B}{\to}DGBialg\stackrel{Q}{\to}DGLieCoalg$$ where $B$ is a bar construction and $Q$ is the functor of indecomposables. The adjoint functor $\mathcal{A}$ is dual to the Chevalley-Eilenberg functor. There is a standard weak equivalence $\mathcal{A}\Gamma(A)\to A$.

I am struggling to find any reference to these functors in the papers cited in the bibliography. For the proof of Theorem 9.1 we seem to need that the functor $\mathcal{A}\Gamma$ applied to the weak equivalences of $C_\infty$-algebras $$\lbrace f_i\rbrace\colon (H(A),\lbrace m_i\rbrace )\to (A,\lbrace d, \mu, 0,\ldots\rbrace)$$ yields a weak equivalence $\mathcal{A}\Gamma(H(A))\to\mathcal{A}\Gamma(A)$ in $CDGAlg$, but this is not stated anywhere and is not obvious to me.

Is the above true, and can anyone explain why?

Where can I read more about the functors $\Gamma$ and $\mathcal{A}$ and their properties, in particular in the setting of $C_\infty$-algebras?