and use functoriality of the monoidal product.

Comodules and right multiplier bicomonads {#sec:tric_comod}
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Our approach to the study of comodules over a counital fusion morphism is based on the use of the following notion.

\[def:left\_multi\_bicomonad\] A [*right multiplier bicomonad*]{} on a monoidal category $\C$ is a functor equipped with natural transformations ${\overrightarrow}G_2:GX.GY\to G(GX.Y)$ and such that the diagrams $$\label{eq:G_check}
\xymatrix@C=20pt{
GX.GY.GZ\ar[r]^-{1.{\overrightarrow}G_2}\ar[d]_-{{\overrightarrow}G_2.1}&
GX.G(GY.Z)\ar[r]^-{{\overrightarrow}G_2}&
G(GX.GY.Z)\ar[d]^-{G({\overrightarrow}G_2.1)}&
GX.GY\ar[r]^-{{\overrightarrow}G_2}\ar[rd]_-{1.\varepsilon}&
G(GX.Y)\ar[d]^-\varepsilon\\
G(GX.Y).GZ
\ar[rr]_-{{\overrightarrow}G_2}&&
G(G(GX.Y).Z)&&
GX.Y}$$ commute for any objects $X,Y,Z$.

A [*left multiplier bicomonad*]{} on $\C$ is a right multiplier bicomonad on the reverse monoidal category $\C\rev$.

To a right multiplier bicomonad $G$, we can associate a natural transformation $$\xymatrix{
GX.GY\ar[r]^-{{\overrightarrow}G_2}&
G(GX.Y)\ar[r]^-{G(\varepsilon.1)}&
G(X.Y).}$$ It satisfies the same associativity condition as the binary part of a monoidal functor $G$ but it has no nullary part; thus it makes $G$ what might be called a [*semimonoidal functor*]{}. A right multiplier bicomonad is indeed a generalization of monoidal comonad (also known as [*‘bicomonad’*]{}) as the following example shows:

\[ex:G\_check\_from\_bicomonad\] Let $(G,\delta,\varepsilon)$ be a monoidal comonad on a monoidal category $\C$; that is, assume the existence of a monoidal structure $(G_2:G(-).G(-)
\to G(-.-), G_0:I\to GI)$ on the functor $G$, and the monoidality of the coassociative comultiplication $\delta:G\to G^2$ and 