Let
C
=
(
C
,
⊗
,
I
)
be a (strict) monoidal category. The center of
C
, also called the Drinfeld center of
C
and denoted
Z
(
C
)
, is the category whose objects are pairs (A,u) consisting of an object A of
C
and a natural isomorphism
u
X
:
A
⊗
X
→
X
⊗
A
satisfying
u
X
⊗
Y
=
(
1
⊗
u
Y
)
(
u
X
⊗
1
)
and
u
I
=
1
A
(this is actually a consequence of the first axiom).
An arrow from (A,u) to (B,v) in
Z
(
C
)
consists of an arrow
f
:
A
→
B
in
C
such that
v
X
(
f
⊗
1
X
)
=
(
1
X
⊗
f
)
u
X
.
The category
Z
(
C
)
becomes a braided monoidal category with the tensor product on objects defined as
(
A
,
u
)
⊗
(
B
,
v
)
=
(
A
⊗
B
,
w
)
where
w
X
=
(
u
X
⊗
1
)
(
1
⊗
v
X
)
, and the obvious braiding .
