Corollary

Let S, S' be two magmas and $g: S \to S'$ a homomorphism. Then there exists one and only one A-algebra homomorphism $u: A^{(S)} \to A^{(S')}$ rendering commutative the diagram

(where the vertical arrows are the canonical mappings).

It suffices to apply Proposition 6 taking $f$ to be the composite mapping $S \xrightarrow{g} S' \to A^{(S')}$.

In particular, if T is a stable subset of the magma S (I, § 1, no. 4), the set of elements $\sum_{s \in T} \alpha_s e_s$ of $A^{(S)}$ is a subalgebra of $A^{(S)}$ canonically isomorphic to the algebra $A^{(T)}$ and sometimes identified with the latter.