Corollary 2

Let $(E_i)_{i \in I}, (F_i)_{i \in I}$ be two finite families of A*-algebras, with* $E_i$ and $F_i$ graded of type $\Delta_i$ for all $i \in I$. For each $i \in I$, let $g_i : E_i \to F_i$ be a graded algebra homomorphism of type $\Delta_i$. Then, if $h_i : E_i \to \varepsilon \bigotimes_{i \in I} E_i$ and $h'_i : F_i \to \varepsilon \bigotimes_{i \in I} F_i$ are the canonical homomorphisms, there exists one and only one homomorphism of graded A*-algebras of type* $\Delta$, $g : \varepsilon \bigotimes_{i \in I} E_i \to \varepsilon \bigotimes_{i \in I} F_i$ such that $g \circ h_i = h'_i \circ g_i$ for all $i \in I$. Also, if each $g_i$ is bijective, so is $g$.

It suffices to apply Corollary 1 to $f_i = h'_i \circ g_i$, noting that conditions (26) then follow from relations (25) applied to the $h'_i$.

The homomorphism defined in Corollary 2 is also denoted by $\varepsilon \bigotimes_i g_i$ (if no confusion can arise); if, for each $i \in I$, $G_i$ is a third graded A-algebra of type $\Delta_i$ and $g'_i : F_i \to G_i$ a graded algebra homomorphism, then

as follows immediately from (30).

In the case of a skew tensor product of graded algebras of type $\mathbf{Z}$, we write $\varepsilon \bigotimes_i f_i$ instead of $\varepsilon \bigotimes_i f_i$ for homomorphisms $f_i : E_i \to F_i$ of graded algebras of type $\mathbf{Z}$; when $I = \{1, 2\}$, this homomorphism is also denoted by $f_1 \varepsilon \otimes f_2$; when $I = \{1, n\}$ and all the $E_i$ (resp. $F_i$) are equal and all the $f_i$ equal to the same homomorphism $f$, we write $\varepsilon f^{\otimes n}$.