Coideal ($\Delta I\subseteq I\otimes C+C\otimes I,\epsilon (I)=0$)

A linear subspace $I$ of a coalgebra satisfying the displayed conditions. These are exactly the conditions for the comultiplication and counit to descend to the quotient vector space $C/I$. An ideal that is also a coideal in a bialgebra gives a quotient bialgebra.