require the existence of a unit and drop all axioms in [@LackStreet:skew] that involve the unit. Then composing $t_1$ with $e.1$, we can still equip $A$ with a (no longer unital but still associative) multiplication $A.A\to A$. But in the absence of a unit, there is no longer a comultiplication $A\to A.A$.

Our study of counital but not necessarily unital fusion morphisms is motivated by Van Daele’s approach to [*(regular) multiplier Hopf algebra*]{} [@VDae:MHA] (over a field) — based on generalizations of the fusion morphisms $t_1$ and $t_2$ (together with $t_3$ and $t_4$ in the regular case) which are in turn counital but non-unital fusion morphisms (in the symmetric monoidal category of vector spaces). In the absence of a unit for the algebra $A$, none of the morphisms $t_1, t_2, t_3$ and $t_4$ determines the others. All of them are needed to formulate the axioms. Although in the absence of a unit none of the fusion morphisms $t_1, t_2, t_3$ and $t_4$ determines a comultiplication $A\to
A.A$; thanks to their compatibility axioms any pair of them determines a generalized comultiplication taking values in the multiplier algebra [@Dauns:multiplier] of $A.A$.

The main aim of this paper is to extend the definition 