Proposition 1

Let M be a magma. Every mapping f of X into M may be extended in a unique way to a morphism of $M(X)$ into M.

By induction on $n \geqslant$, mappings $f_n : M_n(X) \to M$ are defined as follows: let f_1 = f; for n \geq 2, the mapping f_n is defined by $f_n(w.w') = f_p(w).f_{n-p}(w')$ for $p = 1, 2, \ldots, n-1$ and $(w, w')$ in $M_p(X) \times M_{n-p}(X)$. Let $g$ be the mapping of $M(X)$ into $M$ which induces $f_n$ on $M_n(X)$ for every integer $n \geq 1$. Clearly $g$ is the unique morphism of $M(X)$ into $M$ which extends $f$.

Let $u$ be a mapping of $X$ into a set $Y$. By Proposition 1, there exists one and only one homomorphism of $M(X)$ into $M(Y)$ which coincides with $u$ on $X$. It will be denoted by $M(u)$. If $v$ is a mapping of $Y$ into a set $Z$, the homomorphism $M(v) \circ M(u)$ of $M(X)$ into $M(Z)$ coincides with $v \circ u$ on $X$, whence