of [*(regular) multiplier bialgebra*]{} to any braided monoidal category. In formulating the definition, any reference to multipliers is completely avoided. If applying it to the symmetric monoidal category of vector spaces, our definition covers the notion of multiplier bialgebra in [@BoGTLC:wmba Theorem 2.11] but it is slightly more general.

We build up gradually to the full structure of regular multiplier bimonoid, first developing in Section \[sec:counital\_fusion\_morphism\] the theory of a single (counital) fusion morphism $t_1$; then considering in Section \[sec:mbm\] multiplier bimonoids, involving a pair of these, such as $t_1$ and $t_2$ or $t_3$ and $t_4$; before finally considering in Section \[sec:reg\_mbm\] regular multiplier bimonoids, which involve all four fusion morphisms. We observed above that, unlike the situation with bimonoids, the notion of multiplier bimonoid is not stable under passage from a braided monoidal category to its opposite. But there are other duality principles available in braided monoidal category: one can reverse the multiplication, one can replace the braiding by its inverse, or one can do both. The resulting four variants in some sense correspond to the four fusion morphisms in a regular multiplier bimonoid. Our proofs often rely on these duality principles.

At each stage we study the corresponding 