Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian? 
[stated in 1J of “ON THE CENTER OF FUSION CATEGORIES” by Bruiguières and Virelizier]

In particular, I’m not seeing why any monomorphism in the center would have to be a kernel of a morphism? (I’m relatively happy with the other axioms holding, but if anyone has a reference where this is discuss explicitly, it’s appreciated )