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

$$(\lambda_C\otimes id_D)\circ\rho_D = (id_C\otimes \rho_D)\circ \lambda_C.$$

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!