In many places, I have seen the slogan that "simplicial abelian group = chain complexes of abelian groups". These same sources usually tell me how to go in one direction. Namely, given a simplicial abelian group, one gets a chain complex whose $n^{\text{th}}$ term is the abelian group corresponding to the $n$-simplex and whose differential is the alternating sum of the face maps.

However, this process does not appear to me to be reversible. My question then is in what sense is the above slogan true? In other words, is there some procedure that takes a chain complex of abelian groups and produces a simplicial abelian group? If there is, in what sense is it an inverse to the above procedure?