$(1.t')(t'.1)(1.t')=(t'.1)(1.b^{-1})(t'.1)$; an object $A$ equipped with a morphism $A.A\to A.A$ satisfying the cocycle condition is called a lax tricocycloid, or an augmented lax tricocycloid if it is also unital and counital. The existence of an antipode; that is, the Hopf condition on the bimonoid $A$, is equivalent to the invertibility of the fusion morphism $t_1$.

The notion of bimonoid is self-dual; that is, symmetric under reversing the arrows in the diagrams encoding the axioms. However, the above description in terms of the fusion morphism $t_1$ is not self-dual. Instead, there is an equivalent dual description in terms of the morphism $$t_2:=\big(
\xymatrix{
A.A\ar[r]^-{1.d}&
A.A.A\ar[r]^-{m.1}&
A.A}\big)$$ which satisfies a modified fusion equation. In fact, using the braiding one can construct two further variants $$\xymatrix@R=10pt{
t_3:=\big(A.A\ar[r]^-{d.1}&
A.A.A\ar[r]^-{1.b^{-1}}&
A.A.A\ar[r]^-{1.m}&
A.A\big)\\
t_4:=\big(A.A\ar[r]^-{1.d}&
A.A.A\ar[r]^-{b^{-1}.1}&
A.A.A\ar[r]^-{m.1}&
A.A\big)}$$ of the fusion morphism. Each of the morphisms $t_1, t_2, t_3$ and $t_4$ can be expressed in terms of each of the others with the help of the unit and the counit of $A$.

Using the language of fusion morphisms, a (say, right) comodule $V$ over the comonoid $A$ can be described equivalently in terms of a morphism $v:V.A\to
V.A$. Coassociativity and counitality of the coaction $V \to V.A$ 