monoidal categories. This is explained by the structure carried by their induced functors. The definition of (co)modules in terms of [*pairs*]{} of the fusion morphisms $t_1, t_2, t_3$ and $t_4$ goes back to [@VDaZha:corep_I]. This approach becomes particularly important in the generalization [@Bohm:wmba_comod] to [*weak*]{} multiplier bialgebras [@BoGTLC:wmba], when both (co)actions are needed to equip any (co)module with the structure of a bimodule over the so-called base algebra.

One could also define [*multiplier Hopf monoids*]{} in a braided monoidal category as multiplier bimonoids whose constituent fusion morphisms $t_1$ and $t_2$ are isomorphisms. In the symmetric monoidal category of vector spaces this is known to be equivalent to the existence of an [*antipode*]{} taking values in the multiplier algebra [@VDae:MHA]. In our general setting, however, the notion of multipliers is not available. The abstract categorical treatment of multipliers requires the additional assumption that our braided monoidal category is in fact [*closed*]{}. We plan to expound this construction, and the resulting theory of multiplier Hopf monoids, in a subsequent paper [@BL:braided_mha].

Notation {#notation .unnumbered}
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Throughout, $\C$ is a braided monoidal category (unless otherwise stated). We denote the monoidal product by $.$; the monoidal unit by $I$; and the braiding by $b$. 