Proposition 6

Let S be a magma, F an A-algebra and f a homomorphism of

S into F with only its multiplicative structure. Then there exists one and only one A-algebra homomorphism $\bar{f}: A^{(S)} \to F$ rendering commutative the diagram

(where the vertical arrow on the left is the canonical mapping $s \mapsto e_s$).

Let $\bar{f}: A^{(S)} \to F$ be the unique A-module homomorphism such that $\bar{f}(e_s) = \bar{f}(s)$ (II, § 1, no. 11, Corollary 3 to Proposition 17); it suffices to verify that $\bar{f}$ is an algebra homomorphism and for this it suffices to prove that $\bar{f}(e_s e_t) = \bar{f}(e_s) \bar{f}(e_t)$, which follows immediately from the definition and the hypothesis $f(st) = f(s)f(t)$.

Proposition 6 expresses the fact that the ordered pair consisting of $A^{(S)}$ and the canonical mapping $s \mapsto e_s$ is a solution of the universal mapping problem (Set Theory, IV, § 3, no. 1) where $\Sigma$ is the species of A-algebra structure and the $\alpha$-mappings the homomorphisms of S into an A-algebra with only its multiplicative law.