Corollary 1

In the notation of Proposition 10, suppose further that F is a graded A-algebra of type $\Delta$ and that each $f_i$ is a graded algebra homomorphism relative to $\phi_i : \Delta_i \to \Delta$; then $f = \varepsilon \bigotimes_i f_i$ is a graded algebra homomorphism.

This follows immediately from the definition of $f$ and the fact that

by definition of the $\phi_i$.

It is therefore seen that $(E, \psi)$ is also a solution of another universal mapping problem, where this time $\Sigma$ is the species of graded A-algebra structure of type $\Delta$, the morphisms being graded algebra homomorphisms of type $\Delta$ and the $\alpha$-mappings the mappings $\prod_i f_i$, where, in addition to conditions (26), it is assumed that $f_i$ is a graded algebra homomorphism relative to $\phi_i$.