Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be 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 $U,V,W\in\mathcal{C}$
$$J_{U,(W\otimes V)} \circ (U^*\otimes J_{V,W}) \circ \Phi_{U^*,V^*,W^*}=\Phi_{W,V,U}^* \circ J_{(V\otimes U), W}\circ (J_{U,V}\otimes W^*)$$
It should be an easy exercise of diagram chasing, but... I am stuck.