  We generalise the notions of semi-regularity, regularity, and bi-regularity
to unitary solutions of the braided Pentagon equation in concrete W*-categories
with semi-regular/regular/bi-regular braiding, and study their properties. We
show, for example, that under bi-regularity-assumptions, the leg-algebras form
braided C*-bialgebras. Moreover, we "close the circle" for the representation
categories: The braiding on the Yetter-Drinfeld category of a (semi-)regular
multiplicative unitary in a category with (semi-)regular braiding is again
(semi-)regular.
