In Jacob Lurie's paper "Derived Algebraic Geometry," in 2.6 $E_{\infty}$-Ring Spectra and Simplicial Commutative Rings, page 25, there is the following claim:
In general, we have functors $\mathcal{SCR}_{R/}\xrightarrow{\phi}\mathcal{DGA}_{R}\xrightarrow{\psi}\mathcal{EI}_{R/}$. If $R$ is a $\mathbf{Q}$-algebra, then $\psi$ is an equivalence of $\infty$-categories, $\phi$ is fully faithful, and the essential image of $\phi$ consists of the connective objects of $\mathcal{DGA}_{R}\simeq\mathcal{EI}_{R/}$ (that is, those algebras $A$ having $\pi_{i}A=0$ for $i<0$).
What is the explicit functor $\phi:\mathcal{SCR}_{R/}\to\mathcal{DGA}_{R}$? I suppose that the natural thing would be to take a simplicial $R$-algebra $A$ and assign it to \begin{align*} \phi(A)=\bigoplus_{i=0}^{\infty}\pi_{i}A, \end{align*} and take a map $f:A\to B$ and assign it to \begin{align*} \phi(f)=\bigoplus_{i=0}^{\infty}(f_{i}:\pi_{i}A\to\pi_{i}B), \end{align*} but as far as I could find this isn't stated explicitly in DAG. Is this the case, and if so, do you have a source or proof? And how does one show that $\bigoplus\pi_{i}A$ is a differential graded algebra?
The functor SCR→CDGA is given by the normalized chains functor N: sAb→Ch on underlying objects without multiplication, and the multiplication is given by the composition NA⊗NA→N(A⊗A)→NA, where the first map is the Eilenberg–Zilber map and the second map is the normalized chains functor applied to the multiplication map A⊗A→A. The Eilenberg–Zilber map is symmetric, so the above construction indeed lands in commutative differential graded algebras.
See Schwede and Shipley, Equivalences of monoidal model categories for the case of associative algebras. For the commutative case several references can be found at the nLab.