Show that duality functor is anti-monoidal - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T16:49:53Z http://mathoverflow.net/feeds/question/75175 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75175/show-that-duality-functor-is-anti-monoidal Show that duality functor is anti-monoidal Andrea 2011-09-11T23:35:53Z 2011-09-11T23:42:50Z <p>Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let <code>$J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$</code> the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair <code>$((-)^*, J)$</code> is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ <code>$$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$</code> It should be an easy exercise of diagram chasing, but... I am stuck.</p>