Remark

Suppose that the $E_p$ are finitely generated projective $A$-modules, so that the graded $A$-modules $(E \otimes_A E)^{*gr}$ and $E^{*gr} \otimes_A E^{*gr}$ can be canonically identified (II, § 4, no. 4, Corollary 1 to Proposition 4). If also the $A$-modules $A \otimes_A A$ and $A$ are then canonically identified (II, § 3, no. 4), the linear mapping $E^{*gr} \otimes_A E^{*gr} \to E^{*gr}$ which defines multiplication in $E^{*gr}$ can be called the graded transpose of the coproduct $c$.