In general, we have functors ${\mathcal{S}\mathcal{C}\mathcal{R}}_{R/}\stackrel{\varphi}{\to}{\mathcal{D}\mathcal{G}\mathcal{A}}_{R}\stackrel{\psi}{\to}{\mathcal{E}\mathcal{I}}_{R/}$. If $R$ is a $\mathbf{Q}$-algebra, then $\psi $ is an equivalence of $\mathrm{\infty}$-categories, $\varphi $ is fully faithful, and the essential image of $\varphi $ consists of the connective objects of ${\mathcal{D}\mathcal{G}\mathcal{A}}_{R}\simeq {\mathcal{E}\mathcal{I}}_{R/}$ (that is, those algebras $A$ having ${\pi}_{i}A=0$ for $i<0$).

What is the explicit functor $\varphi :{\mathcal{S}\mathcal{C}\mathcal{R}}_{R/}\to {\mathcal{D}\mathcal{G}\mathcal{A}}_{R}$? I suppose that the natural thing would be to take a simplicial $R$-algebra $A$ and assign it to

$\begin{array}{r}\varphi (A)=\underset{i=0}{\overset{\mathrm{\infty}}{\u2a01}}{\pi}_{i}A,\end{array}$

and take a map $f:A\to B$ and assign it to

$\begin{array}{r}\varphi (f)=\underset{i=0}{\overset{\mathrm{\infty}}{\u2a01}}({f}_{i}:{\pi}_{i}A\to {\pi}_{i}B),\end{array}$

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 $\u2a01{\pi}_{i}A$ is a differential graded algebra?