In an article, I encounter an isomorphism relation as follows:

Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism":

$K[x,S]$ isom. to $\Sigma^{-1} Hom_S (K[x,S],S)$, where \Sigma is the translation functor.

I guess this mean we should regard S as a chain complex concentrated at 0, and look at the homomophism between chain complexes, but I cannot figure out the relation.

Explicitly, how to identify the homomophism of chain complexes with K[x,S] as categories?

Thanks a lot.