ct.category theory - Candidate definitions for "1-braided 2-category"?

Recall that a braided monoidal category is a category $mathcal C$, a functor $otimes: mathcal C times mathcal C to mathcal C$ satisfying some properties, and a natural isomorphism $b_{V,W}: Votimes W to Wotimes V$ satisfying some properties. Recall also that a monoidal category (just $mathcal C,otimes$ and their properties) is the same as a one-element 2-category: the objects of $mathcal C$ become the morphisms, and the monoidal structure becomes composition.

Thus, is there a natural definition of "1-braided 2-category"? I'm calling it "1-braided" because the braiding acts on 1-morphisms (as opposed to "braided monoidal 2-category", where the braiding acts on the 0-morphisms).

I realize, of course, that if $V,W$ are morphisms of a 2-category so that $Vcirc W$ is defined, then generally $Wcirc V$ is not defined, so a priori asking for any relationship $Vcirc W cong Wcirc V$ doesn't make sense. On the other hand, consider Aaron Lauda's categorification of $U_q(mathfrak{sl}_2)$. It is a 2-category, but different hom-sets can be more-or-less identified.

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