  We introduce the bicategory of bialgebras with coverings (which can be
thought of as coalgebra-indexed families of morphisms), and provide a
motivating application to the transfer of formulas for primitives and antipode.
Additionally, we study properties of this bicategory and various
sub-bicategories, and describe some universal constructions. Finally, we
generalize Nichols' result on bialgebra quotients of Hopf algebra, which gives
conditions on when the resulting bialgebra quotient is a Hopf algebra.
