of the counit $\varepsilon:G\to 1$. Consider $${\overrightarrow}G_2:=\big(
\xymatrix{
GX.GY\ar[r]^-{\delta.1}&
G^2X.GY\ar[r]^-{G_2}&
G(GX.Y)}\big)\ .$$ Then the diagrams in commute by the coassociativity of $\delta$, the associativity of $G_2$, and by the monoidality of $\delta$ on one hand; and by the monoidality of $\varepsilon$ and the counitality of $\delta$ on the other hand.

Further examples are provided by the functors induced by counital fusion morphisms:

\[ex:G\_check\_from\_t\] Let $(t:A^2\to A^2,e:A\to I)$ be a counital fusion morphism in a braided monoidal category $\C$. Consider the functor $G:=(-).A:\C\to \C$ with the natural transformations $\varepsilon:=1.e$ and $${\overrightarrow}G_2:=\big(
\xymatrix@C=35pt{
X.A.Y.A\ar[r]^-{1.b.1}&
X.Y.A^2\ar[r]^-{1.1.t}&
X.Y.A^2\ar[r]^-{1.b^{-1}.1}&
X.A.Y.A}\big).$$ Then the diagrams in commute by the fusion equation and by the counitality condition on $t$, respectively.

\[def:G\_check\_comodule\] Consider a right multiplier bicomonad $G$ on a monoidal category $\C$. A $G$-[*comodule*]{} is an object $V$ together with a natural transformation ${\overrightarrow}v:V.G(-)\to G(V.-)$ rendering commutative, for any objects $Y$ and $Z$, the diagrams $$\label{eq:v_check}
\xymatrix@C=20pt{
V.GY.GZ\ar[r]^-{1.{\overrightarrow}G_2}\ar[d]_-{{\overrightarrow}v.1}&
V.G(GY.Z)\ar[r]^-{{\overrightarrow}v}&
G(V.GY.Z)\ar[d]^-{G({\overrightarrow}v.1)}&
V.GY\ar[r]^-{{\overrightarrow}v}\ar[rd]_-{1.\varepsilon}&
G(V.Y)\ar[d]^-\varepsilon\\
G(V.Y).GZ\ar[rr]_-{{\overrightarrow}G_2}&&
G(G(V.Y).Z)&&
V.Y.}$$ A [*morphism*]{} of $G$-comodules is a morphism $f:V\to W$ such that $G(f.1){\overrightarrow}v ={\overrightarrow}w (f.1)$.

Comodules over a left multiplier bicomonad are defined as comodules over the corresponding right multiplier bicomonad on $\C\rev$.

These comodules behave well with respect to the monoidal structure of 