Proposition 7

Let I be a set and F an algebra (resp. unital associative algebra, resp. unital commutative associative algebra) over A. For every mapping $f : I \to F$, there exists one and only one homomorphism (resp. unital homomorphism) $\bar{f}$ of LibA(I) (resp. Libas_A(I), resp. Libasc_A(I)) into F such that $\bar{f}(X_i) = f(i)$ for all $i \in I$.*

Let $F_m$ be the magma (resp. monoid) obtained by giving the set F its multiplicative law of composition. There is one and only one homomorphism (resp. unital homomorphism) $g$ of M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) into $F_m$ such that $g(i) = f(i)$ for all $i \in I$ (I, § 7, nos. 1, 2 and 7); Proposition 7 then follows from no. 6, Proposition 6.