nLab
bicomodule

Let $C,D$ be comonoids in a monoidal category $A=(A,\otimes ,1)$. A $C$-$D$ bicomodule is an object $M$ in $A$, with left $C$-coaction ${\lambda }_C:M\to C\otimes M$ and right $D$-coaction ${\rho }_D:M\to M\otimes D$ which commute in the sense that

$$({\lambda }_C\otimes {\mathrm{id}}_D)\circ {\rho }_D=({\mathrm{id}}_C\otimes {\rho }_D)\circ {\lambda }_C.$$(\lambda_C\otimes id_D)\circ\rho_D = (id_C\otimes \rho_D)\circ \lambda_C.

Typical cases are when $A$ is the category of $k$-modules where $k$ is a commutative unital ring (the comonoids are then $k$-coalgebras), and the more general case of bicomodules over corings, where $A$ is the category of $k$-bimodules where $k$ is a possibly noncommutative ring.

There is an operation of cotensor product for bicomodules over coalgebras/corings; however it is not associative in general, unlike the tensor product of bimodules over rings!