  We introduce the notion of a partial corepresentation of a given Hopf algebra
$H$ over a coalgebra $C$ and the closely related concept of a partial
$H$-comodule. We prove that there exists a universal coalgebra $H^{par}$,
associated to the original Hopf algebra $H$, such that the category of regular
partial $H$-comodules is isomorphic to the category of $H^{par}$-comodules. We
introduce the notion of a Hopf coalgebroid and show that the universal
coalgebra $H^{par}$ has the structure of a Hopf coalgebroid over a suitable
coalgebra.
