Cyclic cohomology may be defined in several ways: the easiest way to define it is via a subcomplex $C^*_{\lambda}$of Hochschild complex consisting from cyclic cochains. There are also other definitions for example using the cyclic bicomplex or $(b,B)$ (mixed) complex. Relevant definition can be found here. One can define the so called *periodicity operator* $S:HC^{n}(A) \to HC^{n+2}(A)$ and using this operator one can define the *periodic cyclic cohomology* as the inductive limit of $HC^{2n}(A)$ or $HC^{2n+1}$ (where the arrows of inductive system are just iterations of $S$).

Why is periodic cyclic cohomology of $A$ isomorphic with the cohomology of the totalization of $(b,B)$-bicomplex?

Several remarks:
1. To define *periodic* theory one uses the bicomplex which is no longer concentrated in the first quadrant (see again my previous question linked above).

2. I would be happy if I could see the map yielding quasi-isomorphism explicitly.

3. There are still another variants of cyclic theory such as negative theory for instance. If there is a way to prove that the two approaches: using $(b,B)$ bicomplex and ordinary cyclic complex and $S$ operator are equivalent, without invoking the negative theory, I would be happy to see it.