Monoidal adjunction explained

Suppose that

(lC, ⊗ ,I)

and

(lD,\bullet,J)

are two monoidal categories. A monoidal adjunction between two lax monoidal functors

(F,m):(lC, ⊗ ,I)\to(lD,\bullet,J)

and

(G,n):(lD,\bullet,J)\to(lC, ⊗ ,I)

(F,G,η,\varepsilon)

between the underlying functors, such that the natural transformations

η:1_lC ⇒ G\circF

and

\varepsilon:F\circG ⇒ 1_lD

Lifting adjunctions to monoidal adjunctions

Suppose that

(F,m):(lC, ⊗ ,I)\to(lD,\bullet,J)

is a lax monoidal functor such that the underlying functor

F:lC\tolD

has a right adjoint

G:lD\tolC

. This adjunction lifts to a monoidal adjunction

(F,m)

⊣

(G,n)

if and only if the lax monoidal functor

(F,m)

is strong.