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Suppose that
(
C
,
⊗
,
I
)
{\displaystyle ({\mathcal {C}},\otimes ,I)}
and
(
D
,
∙
,
J
)
{\displaystyle ({\mathcal {D}},\bullet ,J)}
are two monoidal categories . A monoidal adjunction between two lax monoidal functors
(
F
,
m
)
:
(
C
,
⊗
,
I
)
→
(
D
,
∙
,
J
)
{\displaystyle (F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)}
and
(
G
,
n
)
:
(
D
,
∙
,
J
)
→
(
C
,
⊗
,
I
)
{\displaystyle (G,n):({\mathcal {D}},\bullet ,J)\to ({\mathcal {C}},\otimes ,I)}
is an adjunction
(
F
,
G
,
η
,
ε
)
{\displaystyle (F,G,\eta ,\varepsilon )}
between the underlying functors, such that the natural transformations
η
:
1
C
⇒
G
∘
F
{\displaystyle \eta :1_{\mathcal {C}}\Rightarrow G\circ F}
and
ε
:
F
∘
G
⇒
1
D
{\displaystyle \varepsilon :F\circ G\Rightarrow 1_{\mathcal {D}}}
are monoidal natural transformations .
Lifting adjunctions to monoidal adjunctions [ edit ]
Suppose that
(
F
,
m
)
:
(
C
,
⊗
,
I
)
→
(
D
,
∙
,
J
)
{\displaystyle (F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)}
is a lax monoidal functor such that the underlying functor
F
:
C
→
D
{\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}
has a right adjoint
G
:
D
→
C
{\displaystyle G:{\mathcal {D}}\to {\mathcal {C}}}
. This adjunction lifts to a monoidal adjunction
(
F
,
m
)
{\displaystyle (F,m)}
⊣
(
G
,
n
)
{\displaystyle (G,n)}
if and only if the lax monoidal functor
(
F
,
m
)
{\displaystyle (F,m)}
is strong.
Every monoidal adjunction
(
F
,
m
)
{\displaystyle (F,m)}
⊣
(
G
,
n
)
{\displaystyle (G,n)}
defines a monoidal monad
G
∘
F
{\displaystyle G\circ F}
.