situation in Definition \[def:G\_check\_comodule\] are provided by the following.

\[def:tric\_comodule\] Consider a counital fusion morphism $(t:A^2\to A^2,e:A\to I)$ in a braided monoidal category $\C$. A [*comodule*]{} over it is an object $V$ together with a morphism $v:V.A\to V.A$ in $\C$ rendering commutative the following diagrams. $$\label{eq:comod}
\xymatrix{
V.A^2 \ar[r]^-{1.t}\ar[d]_-{v.1}&
V.A^2 \ar[r]^-{b.1}&
A.V.A \ar[r]^-{1.v}&
A.V.A \ar[r]^-{b^{-1}.1}&
V.A^2\ar[d]^-{v.1}&
V.A\ar[r]^-v\ar[rd]_-{1.e}&
V.A\ar[d]^-{1.e}\\
V.A^2 \ar[rrrr]_-{1.t}&&&&
V.A^2&&
V.}$$ A [*morphism*]{} of comodules is a morphism $f:V\to W$ in $\C$ such that $w(f.1)=(f.1)v$.

A comodule $v:V.A\to V.A$ as in Definition \[def:tric\_comodule\] induces a comodule over the right multiplier bicomonad $(-).A$ in Example \[ex:G\_check\_from\_t\] by putting $$\xymatrix{
V.X.A\ar[r]^-{b.1}&
X.V.A\ar[r]^-{1.v}&
X.V.A\ar[r]^-{b^{-1}.1}&
V.X.A.}$$ This is in fact the object-part of a fully faithful functor from the category of comodules for $A$ to the category of comodules for the induced comonad $G$. It’s not hard to see that the functor is injective on objects; in many concrete cases, such as $\C=\vec$, it is an isomorphism of categories.

Hence from Theorem \[thm:mon\_cat\_of\_check\_G\_comodules\] we have the following.

\[cor:mon\_cat\_of\_tric\_comodules\] Consider a counital fusion morphism $(t:A^2\to A^2,e:A\to I)$ in a braided monoidal category $\C$. Its category of comodules, as in Definition \[def:tric\_comodule\], is monoidal in such a way that the evident forgetful functor to $\C$ 