Lemma 2

Let $M$ be the monoidal sum of the family $(M_x)_{x \in X}$ defined by $M_x = \mathbf{N}$ for all $x \in X$ and let $\phi_x$ denote the canonical homomorphism of $M_x$ into $M$. The mapping $x \mapsto \phi_x(1)$ of $X$ into $M$ extends to an isomorphism $h$ of $\mathrm{Mo}(X)$ onto $M$.

Let $h$ be the homomorphism of $\mathrm{Mo}(X)$ into $M$ characterized by $h(x) = \phi_x(1)$. For every integer $n \geq 0$, $\phi_x(n) = \phi_x(1)^n = h(x)^n$ and as $M$ is generated by $\bigcup_{x \in X} \phi_x(\mathbf{N})$, it is also generated by $h(X)$. Hence $h$ is surjective. Moreover, for all $x$ in $X$, the mapping $n \mapsto x^n$ is a homomorphism of $\mathbf{N} = M_x$ into $\mathrm{Mo}(X)$; there thus exists (no. 3, Proposition 4) a homomorphism $h'$ of $M$ into $\mathrm{Mo}(X)$ such that $h'(\phi_x(n)) = x^n$ for $x \in X$ and $n \in \mathbf{N}$; in particular, $h'(h(x)) = x$ for $x \in X$ and hence $h' \circ h$ is the identity homomorphism of $\mathrm{Mo}(X)$. Therefore $h$ is injective. It has thus been proved that $h$ is bijective.