ScalingStacks

Let $g:\{1,\ldots,n\}\to\{1,\ldots,m\}$ be a map and let $r\in\{1,\ldots,m\}$ such that $f^{-1}(r)=\{s\}$ for some $s\in\{1,\ldots,n\}$. Define maps

$$\alpha:\{1,\ldots,n-1\}\to\{1,\ldots,n\},\ i\mapsto\begin{cases}i&\text{ if }i<s\\ i+1&\text{ if }i\geq s\end{cases}$$

and

$$\beta:\{1,\ldots,m\}\to\{1,\ldots,m-1\},\ i\mapsto\begin{cases}i&\text{ if }i<r\\ i-1&\text{ if }i\geq r\end{cases}$$

Then, we have a canonical isomorphism

(1)

$$H^{i}_{d_{r}}(\operatorname{Tot}\nolimits^{f}(M))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Tot}\nolimits^{\beta f\alpha}(H^{i}_{d_{s}}(M)).$$