delooping under Dold-Kan and simplicial delooping

What maps of simplicial sets exist between

• the image under the Dold-Kan correspondence of a chain complex shifted up in degree

• and the image under the right adjoint to simplicial looping of the DK-image of the unshifted complex

?

Here is the same question in detail:

Write

$$(G \dashv \bar W) : sGrp \stackrel{\leftarrow}{\underset{\bar W}{\to}} sSet_0 \hookrightarrow sSet$$

for the adjunction between simplicial groups and reduced simplicial sets whose left adjoint is the simplicial loop group functor (as for instance in Goerss-Jardine, chapter V);

and write

$$Ch_\bullet^+ \overset{\Xi}{\to} sAbGrp \hookrightarrow sGrp \overset{U}{\to} sSet$$

for the Dold-Kan correspondence, where in both cases I care about the images as simplicial sets.

Then for $V \in Ch_\bullet^+$ a chain complex and $V[1]$ (or $V[-1]$ if you prefer) its shift up in degree (its delooping as a chain complex) the two simplicial sets

$$U \Xi (V[1])$$

and

$$\bar W (\Xi V)$$

should have the same homotopy type. What nice natural maps of simplicial sets do we have between them?

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