Exercise 8, § 7

Let $X$ be a set; for every integer $q \geq 0$, let Mo(X)_q be the set of elements of Mo(X) of length q, and let Mo^{(q)}(X) be the union of the Mo(X)_{nq}, for $n \in \mathbf{N}$. The injection Mo(X)_q \to Mo^{(q)}(X) extends to a homomorphism Mo(Mo(X)_q) \to Mo^{(q)}(X); prove that this homomorphism is bijective if $q \geq 1$.