Let be a monoidal category

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

. The centre of , denoted , is the category whose objects are pairs (A,u) consisting of an object A of and a natural isomorphism  satisfying

and

(this is actually a consequence of the first axiom).

An arrow from (A,u) to (B,v) in consists of an arrow in such that
.

The category becomes a braided monoidal category with the tensor product on objects defined as


where , and the obvious braiding .