Proposition 8

(i) The homomorphisms $f_i : E_i \to E = \bigotimes_{k \in I} E_k$ are such that for two indices i, j such that $i \neq j$, $f_i(x_i)$ and $f_j(x_j)$ commute in E for all $x_i \in E_i$ and $x_j \in E_j$; further, E is generated by the union of the subalgebras $f_i(E_i)$.

(ii) Let F be an A-algebra and, for all $i \in I$, let $u_i : E_i \to F$ be an A-algebra homomorphism such that, for $i \neq j$, $u_i(x_i)$ and $u_j(x_j)$ commute in F for all $x_i \in E_i$ and $x_j \in E_j$. Then there exists one and only one A-algebra homomorphism $u : E \to F$ such that $u_i = u \circ f_i$ for all $i \in I$.

(i) As, for every finite subset J of I, $f_i = f_J \circ f_{J,i}$, the first assertion in (i) follows from no. 2, Proposition 5, taking J containing i and j; the second also follows from no. 2, Proposition 5, taking account of the fact that E is the union of the $f_J(E_J)$ when J runs through $\mathscr{F}(I)$.

(ii) For every finite subset J of I, it follows from no. 2, Proposition 5 that there exists a unique homomorphism $u_J : E_J \to F$ such that $u_J \circ f_{J,i} = u_i$ for all $i \in J$; it immediately follows from this uniqueness property that, for $J \subset J'$, $u_J = u_{J'} \circ f_{J'J}$; in other words, the $u_J$ form a direct system of homomorphisms. Let $u = \lim \to u_J : E \to F$; then by definition $u_J = u \circ f_J$ for every finite subset $J$ of $I$ and in particular $u_i = u \circ f_i$ for all $i \in I$; the uniqueness of $u$ follows from these relations and the fact that the $f_i(E_i)$ generate the algebra $E$.