related by the Dold-Kan correspondence
This appears as (Goerss-Jardine, ch V, prop. 6.2).
Under the forgetful functor
into pointed simplicial sets (where the obvious inclusion, and forms the 1st Eilenberg subcomplex over the basepoint) is a Quillen adjunction (with respect to the reduced model structure from prop. 1 and the coslice model structure under the point of the classical model structure on simplicial sets).
By prop. 1 the left adjoint preserves cofibrations and acyclic cofibrations (in fact all weak equivalences).
But by the formula at smash product the reduced suspension clearly lands in reduced simplicial sets, so that we do have the restricted adjunction as claimed. Now by the fact that the classical model structure on simplicial sets is a simplicial model category, the above is a Quillen adjunction and preserves cofibrations and acyclic cofibrations. Hence, by prop. 1, so does its factorization through the model structure on reduced simplicial sets.
The operations of reduced suspension and loop space (co-)restrict to a Quillen adjunction on the reduced model structure (prop. 1) itself, by composition of the Quillen adjunctions from prop. 3 and prop. 4.
This appears as (Goerss-Jardine, ch. V prop. 6.3).
model structure on reduced simplicial sets
A standard textbook reference is chapter V of