Corollary

Let $(E_i)_{i \in I}, (F_i)_{i \in I}$ be two finite families of A-algebras and, for all $i \in I$, let $f_i : E_i \to F_i$ be an algebra homomorphism. If $u_i : E_i \to \bigotimes_{j \in I} E_j, v_i : F_i \to \bigotimes_{j \in I} F_j$ are the canonical homomorphisms, the mapping $f = \bigotimes_i f_i$ (cf. no. 1) is the unique A-algebra homomorphism such that $f \circ u_i = v_i \circ f_i$ for all $i \in I$.

It suffices to note that the homomorphisms $g_i = v_i \circ f_i$ are such that $g_i(x_i) = v_i(f_i(x_i))$ and $g_j(x_j) = v_j(f_j(x_j))$ commute for $i \neq j$, $x_i \in E_i$ and $x_j \in E_j$; then apply Proposition 5.

When, in Proposition 5, the algebra F is assumed to be commutative, the hypothesis that $v_i(x_i)$ and $v_j(x_j)$ are permutable for $i \neq j$ is automatically satisfied. Hence, when F is commutative, there is a canonical bijection

namely the one which associates with every homomorphism $w$ of $\bigotimes_i E_i$ into F the family of $w \circ u_i$.

Note that if E is a commutative A-algebra, the ring structure of $E \otimes_A F$ is the same as that of $F_{(E)}$ (§ 1, no. 5).