I am sorry if this crossposting breaks a rule of MO or SE, but I waited a lot and I hope to get some advices by spamming a little.

Here's the question:

Suppose that you have two sequences $\{F_n\}$ and $\{G_n\}$ of endofunctors of $\bf A$ arising as strings of adjoints $\cdots\dashv F_{n-1}\dashv F_n\dashv F_{n+1}\dashv \cdots$, $\cdots\dashv G_{n-1}\dashv G_n\dashv G_{n+1}\dashv \cdots$.

Then, given a natural transformation $\eta_0\colon F_0\to G_0$, adjointness implies the existence of natural transformations

for any $n\in \mathbb Z$: $\eta_n\colon F_n\to G_n$ is $n$ is even, and $\eta_n\colon G_n\to F_n$ if $n$ is odd. More explicitly, if $n$ is even then I have what I drew above.

One can now consider for any two integers $n\le m$ the composition $$\eta_{n,m}:=\mathop{\boxminus}_{i=n}^m \eta_i \colon \prod_{i=n}^m (F:G)_i \Rightarrow \prod_{i=n}^m (G:F)_i$$ where $\boxminus$ is horizontal composition of natural transformations and $(A:B)$ is a symbol which is equal to $A$ in case $i$ is even, and equal to $B$ if $i$ is odd, and can be defined in the general case as $(A_0:\dots :A_{n-1})_i = A_{j}$ where $i\equiv j\pmod{n}$.

Now I'm wondering if an hypotetical "limit" like $\underline{\eta}\colon \prod_{i=-\infty}^\infty (F:G)_i\Rightarrow \prod_{i=-\infty}^\infty (G:F)_i$ make any sense as a natural transformation between endofunctors of $\bf A$. Has anybody studied a similar problem before? Given some suitable properties of the functors $F_n$, can one say something about $\eta_{n,m}$ and $\underline\eta$?