A normalized 3-cochain gives the gauge equivalence of 2Vect_G^omega

The multiplicative coboundary on a normalized 3-cochain is the five-factor expression in the statement. Its coboundary is $1$: writing $\delta ={\prod }_i{d}_i^{(-1{)}^i}$, where $d_0$ drops the first group element, $d_4$ drops the last, and each interior $d_i$ multiplies adjacent entries, the identities $d_j{d}_i=d_i{d}_{j-1}$ for $i<j$ pair every factor of ${\delta }^2\beta$ with its inverse. The interior identities are exactly associativity in $G$. Hence $\omega \delta \beta$ is a 4-cocycle whenever $\omega$ is. Normalization of $\beta$ makes $\delta \beta$ equal to $1$ if any input is $e$.

Define the underlying pseudofunctor to be the identity on objects, functors, and natural transformations; take its tensorator and unitator to be identity equivalences. For each triple of homogeneous input degrees $(g,h,k)$, define its associativity coherence 2-cell to be multiplication by $\beta (g,h,k)$. Extend to arbitrary objects by direct sums. Since scalar multiplication commutes with every linear map, these components are natural and define an invertible modification.

On four homogeneous inputs, the monoidal 2-functor coherence equation compares the source pentagonator $\omega (g,h,k,l)$, the target pentagonator ${\omega }^{\mathrm{\prime }}(g,h,k,l)$, and the five triple-coherence components. Their ratio is $\beta (h,k,l)\beta (gh,k,l{)}^{-1}\beta (g,hk,l)\beta (g,h,kl{)}^{-1}\beta (g,h,k)=\delta \beta (g,h,k,l)\mathrm{.}$ Thus the coherence equation is precisely ${\omega }^{\mathrm{\prime }}=\omega \delta \beta$. Normalization gives the unit equations, and the same scalar identity on homogeneous summands gives the equation on all direct sums. The underlying functor is the identity, hence an equivalence; its inverse uses ${\beta }^{-1}$.

Conversely, in this gauge family the underlying functor, tensorator, and unitator are fixed as identities. Any normalized scalar triple-coherence modification is a normalized 3-cochain $\beta$, and its four-input coherence forces ${\omega }^{\mathrm{\prime }}\mathrm{/}\omega =\delta \beta$ by step 2.1. Therefore $\omega$ can be untwisted by these data exactly when it is a coboundary, equivalently when $[\omega ]=0$. This makes no claim about equivalences that change the grading group or the underlying 2-category.