# 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.
{"ft_lang_label":"__label__en","ft_lang_prob":0.64430606,"math_prob":0.9998179,"size":565,"snap":"2020-10-2020-16","text_gpt3_token_len":204,"char_repetition_ratio":0.12477718,"word_repetition_ratio":0.0,"special_character_ratio":0.3415929,"punctuation_ratio":0.13934426,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99987423,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-29T04:11:29Z\",\"WARC-Record-ID\":\"<urn:uuid:cc448a68-4405-4786-9c35-b5530caf8801>\",\"Content-Length\":\"111351\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:548864b6-080b-4fd8-9e49-5777accb0b05>\",\"WARC-Concurrent-To\":\"<urn:uuid:4dfb6ea6-5e9f-4b42-a8d6-96cce3e0e7e0>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/75175/show-that-duality-functor-is-anti-monoidal\",\"WARC-Payload-Digest\":\"sha1:Q6NIPP54L33Q65OB2XTORNA7ZLLVS7QV\",\"WARC-Block-Digest\":\"sha1:DVMGDNRLHFZCU47FLAWFQ52AGVBTO5EJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370493684.2_warc_CC-MAIN-20200329015008-20200329045008-00274.warc.gz\"}"}