the functors induced by a counital fusion morphism.

Definition and properties {#sec:counital_fusion_morphism}
-------------------------

Lax tricocycloids — corresponding bijectively to fusion morphisms — were defined in [@LackStreet:skew]; where they were equipped both with a counit and a unit. We need the following weakening.

\[def:counital\_tricocycloid\] A [*counital fusion morphism*]{} in a braided monoidal category $\C$ is given by a pair of morphisms $t:A^2\to A^2$ (called a [*fusion morphism*]{}) and $e:A\to I$ (called the [*counit*]{}) such that the following diagrams commute. $$\label{eq:counital_tricoc}
\xymatrix{
A^3 \ar[r]^-{1.t}\ar[d]_-{t.1}&
A^3 \ar[r]^-{b.1}&
A^3 \ar[r]^-{1.t}&
A^3 \ar[r]^-{b^{-1}.1}&
A^3\ar[d]^-{t.1}&
A^2\ar[r]^-t\ar[rd]_-{1.e}&
A^2\ar[d]^-{1.e}\\
A^3 \ar[rrrr]_-{1.t}&&&&
A^3&&
A}$$ We refer to the first condition as the [*fusion equation*]{} and to the second one as the [*counitality condition*]{}.

A class of examples, albeit unital ones, comes from bimonoids in braided monoidal categories:

\[ex:bimonoid\] Consider a bimonoid $A$ in a braided monoidal category $\C$. Denote its monoid structure by $(m:A^2\to A, u:I\to A)$ and denote the comonoid structure by $(d:A\to A^2, e:A\to I)$. Then $$t:=\big(
\xymatrix{
A^2\ar[r]^-{d.1}&
A^3\ar[r]^-{1.m}&
A^2}\big)$$ is a counital fusion morphism. This is easiest to see using string diagrams; when we draw

$$\begin{tikzpicture} \path (0,1) node[empty,name=s1t] {}
(1,1) node[empty,name=s2t]  {} 
(0.5,0.5) node[empty,name=m] {} 
(0.5,0) node[empty,name=sb] {}
(1.5,0.5) node{} ;
\path (-0.5,0.5) node 