# Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

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