Homomorphism from tensor power of Abelianization to lower central series factor

Definition
Let $$G$$ be a group, and let $$\gamma_n(G)$$ denote the lower central series of $$G$$. In other words, $$\gamma_1(G) = G$$, and $$\gamma_{n+1}(G) = [\gamma_n(G), G]$$. Then, if $$A$$ denotes the Abelianization of $$G$$, there is, for every natural number $$n$$, a homomorphism from $$A^{\otimes n}$$ to $$\gamma_n(G)/\gamma_{n+1}(G)$$, given as follows on the decomposable tensor:

$$(a_1 \otimes a_2 \otimes \dots \otimes a_n) = [[\dots [ g_1, g_2], g_3], \dots, g_n]\gamma_{n+1}(G)$$

where $$g_i$$ is chosen as an arbitrary element in $$G$$ that maps to $$a_i$$ under the Abelianization map $$G \to G/G' = A$$.