We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration.
\begin{lemma}
$A(i) : A(\Delta[n])\to A(\del\Delta[n])$ is surjective.
\end{lemma}
\begin{proof}
Let $\phi\in A(\del\Delta[n])$ be an element of degree $k$, hence it is a map $\del\Delta[n]\to\Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is Kan and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
\cimage[scale=0.5]{Extend_Boundary_Form}
\end{proof}
\begin{lemma}
$A(j) : A(\Delta[n])\to A(\Lambda^n_k)$ is surjective and a quasi isomorphism.
\end{lemma}
\begin{proof}
As above we get surjectivity from the Kan condition. To prove that $A(j)$ is a quasi isomorphism we pass to the singular cochain complex and use that $C^\ast(j) : C^\ast(\Delta[n])\we C^\ast(\Lambda^n_k)$ is a quasi isomorphism. Consider the following diagram and conclude that $A(j)$ is surjective and a quasi isomorphism.
\cimage[scale=0.5]{A_Preserves_WCof}
\end{proof}
Since $A$ is a left adjoint, it preserves all colimits and by functoriality it preserves retracts. From this we can conclude the following corollary.
\begin{corollary}
$A$ preserves all cofibrations and all trivial cofibrations and hence is a left Quillen functor.
\end{corollary}
\begin{corollary}
$A$ and $K$ induce an adjunction on the homotopy categories:
