is strict monoidal.

It suffices to observe that the full subcategory of $G$-comodules consisting of the comodules for $A$ is closed under the monoidal structure. Explicitly, the monoidal unit $I$ is a comodule via the identity morphism $A\to A$ and the monoidal product of comodules $v:V.A\to V.A$ and $w:W.A \to W.A$ is a comodule via $$\xymatrix@C=30pt{
V.W.A\ar[r]^-{1.w}&
V.W.A\ar[r]^-{b.1}&
W.V.A\ar[r]^-{1.v}&
W.V.A\ar[r]^-{b^{-1}.1}&
V.W.A.}$$

Modules and left multiplier bimonads {#sec:tric_module}
------------------------------------

In studying modules over a counital fusion morphism, we rely on the following notion.

\[def:right\_multi\_bimonad\] A [*left multiplier bimonad*]{} on a monoidal category $\C$ is a functor $T:\C\to \C$ equipped with a natural transformation ${\overleftarrow}T_2: T(X.TY)
\to TX.TY$ and a morphism $T_0:TI\to I$ such that the diagrams $$\label{eq:T_hat}
\xymatrix@C=20pt{
T(X.T(Y.TZ))\ar[r]^-{T(1.{\overleftarrow}T_2)}\ar[d]_-{{\overleftarrow}T_2}&
T(X.TY.TZ) \ar[r]^-{{\overleftarrow}T_2}&
T(X.TY).TZ\ar[d]^-{{\overleftarrow}T_2.1}&
T(X.TI)\ar[r]^-{{\overleftarrow}T_2}\ar[rd]_(.36){T(1.T_0)}&
TX.TI\ar[d]^(.56){1.T_0}\\
TX.T(Y.TZ)\ar[rr]_-{1.{\overleftarrow}T_2}&&
TX.TY.TZ&&
TX}$$ commute for any objects $X,Y,Z$.

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

To a left multiplier bimonad $T$, we can associate a natural transformation $$\label{eq:mu_on_T}
\mu:=\big(
\xymatrix{
T^2X\ar[r]^-{{\overleftarrow}T_2}&
TI.TX\ar[r]^-{T_0.1}&
TX}\big).$$ This yields an associative but non-unital multiplication. Left multiplier bimonads are indeed a generalization of opmonoidal monads (also known as [*‘bimonads’*]{}) as the following example shows:

\[ex:T\_hat\_from\_bimonad\] Let $(T,\mu,\eta)$ be an opmonoidal monad on 