# Monoidal equivalences

Consider the strict 2-category of monoidal categories with (lax) monoidal functors as 1-morphisms. There is a general notion of adjunctions and (adjoint) equivalences in (strict) 2-categories. Call these monoidal adjunctions resp. monoidal equivalences in our special case.

Let $L\vdash R$ be a monoidal equivalence. Is it true that $L$ and $R$ are strong monoidal? Note that there is an oplax structure on $F$ given by $$L(X\otimes Y)\longrightarrow L(RLX\otimes RLY)\longrightarrow LR(LX\otimes LY)\longrightarrow LX\otimes LY$$ (in fact, this only requires $L$ to be monoidally left adjoint to $R$), and similarly for $R$.

But I am not able to prove that this is inverse to the lax structure morphism of $L$. Is it even true?

• 1) You mean that $F,G$ are strong monoidal, right? 2) I assume that you already know that $F,G$ is part of a monoidal adjunction. Have you tried to use that unit and counit are monoidal transformations? – HeinrichD Apr 14 '17 at 16:54
• 1) Yes, you are right. – Jakob Werner Apr 14 '17 at 17:03

It is indeed true that a monoidal adjunction suffices, i.e. the left adjoint of a lax monoidal adjunction is necessarily strong. It seems this also follows from a more general principle called doctrinal adjunction invented by Kelly, but a direct verification is also possible. The diagram pasting goes as follows: Here $\varphi$ denotes both $LX\otimes LY\longrightarrow L(X\otimes Y)$ and $RX\otimes RY\longrightarrow R(X\otimes Y)$.