  For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result
states that $A$ is a right $L$-comodule algebra if and only if $A$ is an
algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the
former notion is formulaic while the latter is categorical. We generalize this
result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$
admits a monoidal structure by work of Nill and B\"{o}hm-Caenepeel-Janssen, but
the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras.
Nevertheless, we prove that there is an isomorphism between the category of
right $H$-comodule algebras and the category of algebras in $\mathcal{M}^H$. We
also recall and introduce the formulaic notion of $H$ coacting on a
$\Bbbk$-coalgebra and on a Frobenius $\Bbbk$-algebra, respectively, and prove
analogous category isomorphism results. Our work is inspired by the physical
applications of Frobenius algebras in tensor categories and by symmetries of
algebras with a base algebra larger than the ground field (e.g. path algebras).
We produce examples of the latter by constructing a monoidal functor from a
certain corepresentation category of a bialgebra $L$ to the corepresentation
category of a weak bialgebra built from $L$ (a "quantum transformation
groupoid"), thereby creating weak quantum symmetries from ordinary quantum
symmetries.
