Orthogonal direct sum of cocycles across acting groups

Definition

Suppose $m$ and $n$ are natural numbers. Consider a bunch of groups $G_1, G_2, \dots, G_m$, all equipped with actions on an abelian group $A$. Suppose, for each $i$, that we have a $n$-cocycle $c_i: G_i^n \to A$ for the action of $G_i$ on $A$.

Then, consider $G := G_1 \times G_2 \times \dots \times G_m$. If all the $G_i$-actions on $A$ commute with each other, then we get an induced $G$-action on $A$. We can define a $n$-cocycle $c$ for this action in terms of $c_i$s, as the orthogonal direct sum of cocycles as follows:

First, view $G$ as an internal direct product, so each $G_i$ is identified with a subgroup of $G$. Now, define:

$c((g_{11},g_{12},\dots, g_{1m}),(g_{21},g_{22},\dots,g_{2m}), \dots,(g_{n1},g_{n2},\dots,g_{nm}) = \sum_{i=1}^m c_i(g_{1i},g_{2i},\dots,g_{ni})$

where the summation on the right happens in $A$.

This does indeed give a cocycle.