Let $X$ be a simplicial abelian group. Let $NX$ be its normalised chain complex denoted 


...$\rightarrow NX\_{K}$ $\rightarrow$ $NX\_{k-1}$ $\rightarrow$...


Now define a new chain complex $Y$ by shifting the normalised chain complex; i.e. $Y\_{k} = NX\_{k+1}$. Now applying the functor from Dold Kan correspondence on $Y$ we get a simplicial abelian group $\Gamma(Y)$ whose homotopy groups are isomorphic to those of $\Omega X$, since simplical abelian groups are cofibrant-fibrant one can say $\Gamma(Y)$ is isomorphic to $\Omega X$ in homotopy category (using Whitehead's theorem) provided there is a map $\Omega X$ $\rightarrow$ $\Gamma(Y)$ inducing an isomorphism of homotopy groups.


My question is , is there such a canonical map $\Omega X$ $\rightarrow$ $\Gamma(Y)$ .



Loop space of a Simplicial Abelian group
You can find such a map, but it goes the other way: $\Gamma(Y)\to \Omega X$. It is always wise to keep one's right adjoints on the right hand side.t


But first, let me note that every simplicial abelian group $Z$ is equivalent to $\prod\_{k\ge0} K(\pi\_kZ,k)$ (this is just a restatement of the classical fact that every chain complex of $\mathbb{Z}$-modules is quasi-isomorphic to its homology), so just the observation about homotopy groups suffices to conclude that $\Gamma(Y)$ and $\Omega X$ are equivalent.


Now, for the construction of the map, note that there is a canonical diagram
$$\require{AMScd}
\begin{CD}
 Y @>>> 0\\
 @VVV @VVV\\
 0 @>>> N(X)
\end{CD}
$$
equipped with a chain homotopy of the zero map $Y\to N(X)$ with itself (the latter is just the canonical isomorphism $Y[1]\to N(X)$).


Since Dold-Kan sends chain homotopies to simplicial homotopies, after applying $\Gamma$ we get a homotopy commutative square 
$$\require{AMScd}
\begin{CD}
 \Gamma(Y) @>>> \*\\
 @VVV @VVV\\
 \* @>>> X
\end{CD}
$$
which again induces the required map from $\Gamma(Y)$ to the homotopy pullback of the rest of the diagram, that is $\Omega X$.


There is a more concrete (although messier) way of obtaining the above map, and it is via constructing the adjoint map $\Sigma \Gamma(Y)\to X$. This is because you can get the suspension via the **Kan suspension** functor ([1], III.5)
$$(\Sigma K)\_n = K\_n \vee K\_{n-1}\vee \cdots \vee K\_0$$
so to get map $\Sigma \Gamma(Y)\to X$ you just need to give maps
$$(\Gamma(Y))\_l=\bigoplus\_{[l]\twoheadrightarrow [i]} Y\_i = \bigoplus\_{[l]\twoheadrightarrow [i]} (NX)\_{i+1}\to X\_n =\bigoplus\_{[n]\twoheadrightarrow [i]} (NX)\_{i}$$
for every $l\le n$ satisfying some compatibilities. Spoiler alert: these will just be the canonical inclusions.


In fact what we are doing is just identifying $X$ with $\bar W \Gamma(Y)$, where $\bar W$ is the Eilenberg-MacLane classifying space for a simplicial group ([1] Remark III.5.6).


[1] Goerss, Jardine *Simplcial Homotopy Theory*, 2006


