coalgebras. However, the degrees of comultiplications ${{\Delta}}_{{{\mathfrak{c}}}}$ and ${{\Delta}}_{{{\mathfrak{o}}}}$ are different. This is why $\rho$ is forced to have degree $-1$.

Composing the comultiplication ${{\Delta}}_{{{\mathfrak{o}}}}$ with $\rho \otimes {{\rm id}}$ and ${{\rm id}}\otimes \rho$ we get the maps $$\label{mu-left}
\mu_l = \rho \otimes {{\rm id}}\circ {{\Delta}}_{{{\mathfrak{o}}}} : A \to V\otimes A\,,$$ and $$\label{mu-right}
\mu_r = {{\rm id}}\otimes \rho \circ {{\Delta}}_{{{\mathfrak{o}}}} : A \to A \otimes V\,,$$ respectively.

Using the axioms of ${{\bf S}}$-coalgebras it is not hard to show that

\[prop:comodules\] If we view $V$ as a coassociative coalgebra then the maps $\mu_l$ (\[mu-left\]) and $\mu_r$ (\[mu-right\]) are left and right comodule structures on $A$ over $V$, respectively.

Axiom (\[left-right-via-rho\]) implies that the comodule structure $\mu_r$ is obtained from $\mu_l$ via switching the tensor factors.

To prove that $\mu_l$ is a left comodule structure we need to show that the diagram $$\label{mu-l-comodule}
\xymatrix@M=0.4pc{
A \ar[d]_{\mu_l} \ar[r]^{\mu_l}  &   V \otimes A \ar[d]^{{{\Delta}}_{{{\mathfrak{c}}}} \otimes {{\rm id}}} \\
V \otimes A \ar[r]^-{{{\rm id}}\otimes \mu_l} & V \otimes V \otimes A
}$$ commutes.

The latter follows easily from the commutativity of the diagram below: $$\label{mu-l-comodule-proof}
\xymatrix@M=0.6pc{
A \ar[d]_{{{\Delta}}_{{{\mathfrak{o}}}}} \ar[r]^{{{\Delta}}_{{{\mathfrak{o}}}}} &  A\otimes A \ar[r]^{\rho\otimes 1}
\ar[d]^{{{\Delta}}_{{{\mathfrak{o}}}} \otimes 1}  &       V \otimes A\ar[d]^{{{\Delta}}_{{{\mathfrak{c}}}} \otimes 1} \\
