There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA}\to\CDGA$) and we want to extend to a contravariant functor $\sSet\to\CDGA_\k$. This will be done via \Def{Kan extensions}.
A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial. Secondly we see readily that $F^\ast$ is functorial. By using the definition of colimit and the Yoneda lemma (Y) we can prove that $F_!$ is left adjoint to $F^\ast$:
In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description:
where the addition, multiplication and differential are defined pointwise. Conclude that we have the following contravariant functors (which form an adjoint pair):
\begin{align*}
A(X) &= \Hom_\sSet(X, \Apl) & X \in\sSet\\
K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in\CDGA_\k.
\end{align*}
\subsection{The singular cochain complex}
Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's:
$$ C_n = C^\ast(\Delta^n; \k). $$
The inclusion maps $d^i : \Delta^n \to\Delta^{n+1}$ and the maps $s^i: \Delta^n \to\Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions
\cimage[scale=0.5]{C_Extension}
where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X)\we C^\ast(X)$ for every $X \in\sSet$. Furthermore this map preserves multiplication on the homology algebras.