Several books on homological algebra state that the Künneth spectral sequence specializes to the short exact sequence of the Künneth formula under suitable conditions. My question is whether there is a reference which derives the maps in the Künneth formula explicitely using only the machinery of spectral sequences.
In more detail, I am mainly interested in the following:
Let $R$ be a ring, $F_\bullet$ be a right-bounded complex of flat right $R$-modules and $M$ a left $R$-module. There is a convergent spectral sequence, the Künneth spectral sequence for Homology, $$ E_{pq}^2= Tor^R_p(H_q(F_\bullet), M) \Rightarrow H_{p+q}(F_\bullet \otimes_R M).$$
Since the complex $F_\bullet$ is right-bounded, there is some integer $n \in \mathbb{Z}$ such that $H_n(F_\bullet) \neq 0 = H_i(F_\bullet)$ for any smaller integer $i \leq n$. In other words, $n$ denotes the minimal degree at which the homology of $F_\bullet$ does not vanish. At this degree the Künneth spectral sequence delivers some isomorphism $$\kappa: H_n(F_\bullet) \otimes_{R} M \cong H_n(F_\bullet \otimes_{R} M).$$
My question is whether it is hard to see using the spectral sequence above that $\kappa$ is the pleasant map $$ (x + im \, d_{n-1}) \otimes y \longmapsto x \otimes y + im\, (d_{n-1} \otimes id_M) $$ for any $x \in \ker d_n$ and $y \in M$?
Unfortunately, I could not find a reference deducing the form of $\kappa$ from the spectral sequence alone.
What I could find was that $\kappa$ is born from some diagram chasing when proving the Künneth formula by which I mean the exactness of the sequence $$ 0 \to H_n(F_\bullet) \otimes_{R} M \overset{\kappa}{\to} H_n(F_\bullet \otimes_{R} M) \to Tor_1^R(H_{n-1}(F_\bullet), M) \to 0 $$ in the case that also the boundaries of $F_\bullet$ are flat.
Any comments will be very much appreciated!