Given a commutative unital ring and a morphism of -coalgebras, one can consider the dualized notion of induced module, using the cotensor product instead of tensor product.
If is flat as a -module (e.g. is a field), is a left - right -bicomodule and is a left -comodule, then the cotensor product is a -subcomodule of . In particular, under the flatness assumption, if is a surjection of coalgebras then is a left - right -bicomodule via and respectively, hence is a functor from left - to left -comodules called the induction functor for left comodules from to .
One can consider this construction more generally for corings.