nLab
induced comodule

Given a commutative unital ring $k$ and a morphism $D\to C$ of $k$-coalgebras, one can consider the dualized notion of induced module, using the cotensor product instead of tensor product.

If $D$ is flat as a $k$-module (e.g. $k$ is a field), and $N$ a left $D$- right $C$-bicomodule, then the cotensor product $N\square M$ is a $D$-subcomodule of $N{\otimes }_{k}M$. In particular, under the flatness assumption, if $\pi :D\to C$ is a surjection of coalgebras then $D$ is a left $D$- right $C$-bicomodule via ${\Delta }_D$ and $(id\otimes \pi )\circ {\Delta }_D$ respectively, hence ${\mathrm{Ind}}_C^D:=D{\square }^C-$ is a functor from left $C$- to left $D$-comodules called the induction functor for left comodules from $C$ to $D$.

One can consider this construction more generally for corings.