  Quantum categories have been recently studied because of their relation to
bialgebroids, small categories, and skew monoidales. This is the first of a
series of papers based on the author's PhD thesis in which we examine the
theory of quantum categories developed by Day, Lack, and Street.
  A quantum category is an opmonoidal monad on the monoidale associated to a
biduality $R\dashv R^{\circ}$, or enveloping monoidale, in a monoidal
bicategory of modules $\mathsf{Mod}(\mathcal{V})$ for a monoidal category
$\mathcal{V}$. Lack and Street proved that quantum categories are in
equivalence with right skew monoidales whose unit has a right adjoint in
$\mathsf{Mod}(\mathcal{V})$. Our first important result is similar to that of
Lack and Street. It is a characterisation of opmonoidal \emph{arrows} on
enveloping monoidales in terms of a new structure named \emph{oplax action}. We
then provide three different notions of comodule for an opmonoidal arrow, and
using a similar technique we prove that they are equivalent. Finally, when the
opmonoidal arrow is an opmonoidal monad, we are able to provide the category of
comodules for a quantum category with a monoidal structure such that the
forgetful functor is monoidal.
