The functors Wbar and Diag o Nerve are simplicially homotopy equivalent

Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that Wbar G is a strong simplicial deformation retract of Diag N G. In particular, Wbar G and Diag N G are simplicially homotopy equivalent.