A monoid is given by an object A and arrows m and eta for multiplication and unit. The braiding enters in considering AoB to be a monoid. Dually, a comonoid has a comultiplication and counit. A bimonoid is a monoid A which also has a comonoid structure. The trick to defining a weak bimonoid is to carefully choose a self dual set of diagrams such that bimonoids are always weak bimonoids.
The aim is to relate this definition to a concept of quantum category. In this setting the term quantum involves a linearisation process (so this is not really a quantum gravity kind of quantum). A category is usually specified by source and target maps from C1 to C0, the arrow and object sets. On linearisation one moves into a category of vector spaces rather than sets, and the objects C1 and C0 are comonoids in this category.
How about working with more general monoidal categories? A quantum category is by definition such a category V, with maps s and t into opp(C0) and C0 respectively. Note that the condition of oppositeness is null in the case of vector spaces. This is a natural definition because it looks like the dual of a diagram defining a bialgebroid (I looked on Wiki but they haven't quite got this far). In this dual diagram we consider objects A and R, where A is a weak bimonoid and R is the analogue of C0.
The nice diagrams allow one to show that R can be recovered from the structure of A. The proof crucially uses the idempotents t and s, and the splitting of idempotents in the sense of a Karoubi envelope.