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Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) \otimes_{\mathcal D} F(Y) \to F(X \otimes_{\mathcal C} Y)$, the latter natural in $X,Y\in \mathcal C$, compatible with associators and unitors. It is oplax monoidal if it is instead equipped with maps in the other direction, and strong monoidal if it is equipped with isomorphisms. For each choice of lax/oplax/strong, there is a bicategory of monoidal categories, lax/oplax/strong monoidal functors, and monoidal natural transformations.

Suppose that $F : \mathcal C \to \mathcal D$ is one of lax/oplax/strong monoidal, and also admits a left adjoint $F^L$ in the bicategory of all functors. Under what circumstances is $F^L$ naturally lax/oplax/strong monoidal? Under what circumstances does this adjunction come from an adjunction in the bicategory of monoidal categories and lax/oplax/strong monoidal functors?

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  • $\begingroup$ Of course, the answer will probably be different for different ordered pairs in {lax,oplax,strong}. For example, if $F$ is lax, it is not too hard to believe that $F^L$ is oplax, based on the way that adjoints work (e.g. we do have a canonical map $F^L(1_D) \to 1_C$ corresponding to $1_D \to F(1_C)$). $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 21, 2015 at 22:24
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    $\begingroup$ If $F$ is lax $F^L = L$ is oplax via $L1 \rightarrow 1$ that you referred to, and the $L(X\otimes Y) \rightarrow L(FLX\otimes FLY) \rightarrow LF(LX\otimes LY) \rightarrow LX\otimes LY$. Nothing else is granted. In particular, if $F$ is strong, then $L$ is only oplax, and this is necessarily the only situation when we have an adjoint pair of oplax functors. $\endgroup$
    – Dimitri Chikhladze
    Commented Jul 21, 2015 at 23:08
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    $\begingroup$ This is a old standard question, this is generalizable also to (op)lax.functors (see a bicategory as a monoidal with many objects). Do you read Italian? (I wrote about..) $\endgroup$
    – Buschi Sergio
    Commented Jul 21, 2015 at 23:38

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If $L$ and $R$ are a left and right adjoint, then doctrinal adjunction asserts that $L$ is oplax monoidal iff $R$ is lax monoidal. (I'm being a bit imprecise here, treating monoidality as if it were a property instead of a structure, but hopefully the meaning is clear.)

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    $\begingroup$ Doctrinal adjunction also answers the second question: if $R$ is lax, then its left adjoint $L$ is automatically oplax, and the whole adjunction lifts to an adjunction in the 2-category of lax monoidal functors iff $L$ is actually strong (i.e. its canonically induced oplax structure maps are invertible). $\endgroup$
    – Mike Shulman
    Commented Jul 30, 2015 at 4:23

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