Monoidal adjunction

Suppose that [math]\displaystyle{ (\mathcal C,\otimes,I) }[/math] and [math]\displaystyle{ (\mathcal D,\bullet,J) }[/math] are two monoidal categories. A monoidal adjunction between two lax monoidal functors

[math]\displaystyle{ (F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) }[/math] and [math]\displaystyle{ (G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I) }[/math]

is an adjunction [math]\displaystyle{ (F,G,\eta,\varepsilon) }[/math] between the underlying functors, such that the natural transformations

[math]\displaystyle{ \eta:1_{\mathcal C}\Rightarrow G\circ F }[/math] and [math]\displaystyle{ \varepsilon:F\circ G\Rightarrow 1_{\mathcal D} }[/math]

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

[math]\displaystyle{ (F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) }[/math]

is a lax monoidal functor such that the underlying functor [math]\displaystyle{ F:\mathcal C\to\mathcal D }[/math] has a right adjoint [math]\displaystyle{ G:\mathcal D\to\mathcal C }[/math]. This adjunction lifts to a monoidal adjunction [math]\displaystyle{ (F,m) }[/math]⊣[math]\displaystyle{ (G,n) }[/math] if and only if the lax monoidal functor [math]\displaystyle{ (F,m) }[/math] is strong.

See also

• Every monoidal adjunction [math]\displaystyle{ (F,m) }[/math]⊣[math]\displaystyle{ (G,n) }[/math] defines a monoidal monad [math]\displaystyle{ G\circ F }[/math].