Suppose that
(
C
,
⊗
,
I
)
and
(
D
,
∙
,
J
)
are two monoidal categories. A monoidal adjunction between two lax monoidal functors
(
F
,
m
)
:
(
C
,
⊗
,
I
)
→
(
D
,
∙
,
J
)
and
(
G
,
n
)
:
(
D
,
∙
,
J
)
→
(
C
,
⊗
,
I
)
is an adjunction
(
F
,
G
,
η
,
ε
)
between the underlying functors, such that the natural transformations
η
:
1
C
⇒
G
∘
F
and
ε
:
F
∘
G
⇒
1
D
are monoidal natural transformations.
Suppose that
(
F
,
m
)
:
(
C
,
⊗
,
I
)
→
(
D
,
∙
,
J
)
is a lax monoidal functor such that the underlying functor
F
:
C
→
D
has a right adjoint
G
:
D
→
C
. This adjuction lifts to a monoidal adjuction
(
F
,
m
)
⊣
(
G
,
n
)
if and only if the lax monoidal functor
(
F
,
m
)
is strong.
