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.

Proof that the definition works

 * Orthogonal direct sum of cocycles is cocycle