a monoidal category $\C$; that is, assume the existence of an opmonoidal structure $(T_2:
T(-.-)\to T(-).T(-), T_0:TI\to I)$ and the opmonoidality of the associative multiplication $\mu:T^2\to T$ and of the unit $\eta:1\to T$. Consider $${\overleftarrow}T_2:=\big(
\xymatrix{
T(X.TY)\ar[r]^-{T_2}&
TX.T^2Y\ar[r]^-{1.\mu}&
TX.TY
}\big) .$$ Then the diagrams in commute by the associativity of $\mu$, the coassociativity of $T_2$, and the opmonoidality of $\mu$ on one hand, and by the opmonoidality of $\mu$ and the counitality of $T_2$ on the other hand.

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

\[ex:T\_hat\_from\_t\] 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 $T:=A.(-):\C\to \C$ with the morphism $T_0:=e:A\to I$ and the natural transformation $${\overleftarrow}T_2:=\big(
\xymatrix@C=35pt{
A.X.A.Y\ar[r]^-{1.b^{-1}.1}&
A.A.X.Y\ar[r]^-{t.1.1}&
A.A.X.Y\ar[r]^-{1.b.1}&
A.X.A.Y}\big).$$ Then the diagrams in commute by the fusion equation and by the counitality condition on $t$, respectively. In this example, the multiplication is induced by the multiplication in Proposition \[prop:multiplication\] (1).

\[def:T\_hat\_module\] Consider a left multiplier bimonad $T$ on a monoidal category $\C$. A $T$-[*module*]{} is an object $Q$ together with a natural transformation ${\overleftarrow}q:
T(-.Q)\to T(-).Q$ such that the diagram $$\xymatrix{
T(X.T(Y.Q)) \ar[r]^-{T(1.{\overleftarrow}q)}\ar[d]_-{{\overleftarrow}T_2}&
T(X.TY.Q)\ar[r]^-{{\overleftarrow}q}&
T(X.TY).Q\ar[d]^-{{\overleftarrow}T_2.1}\\
TX.T(Y.Q)\ar[rr]_-{1.{\overleftarrow}q}&&
TX.TY.Q}$$ commutes for any objects $X$ 