Remark 4.6.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ and $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ be the opposite morphisms. Then we have a canonical isomorphism of simplicial sets
\[ (\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq (\operatorname{\mathcal{D}}^{\operatorname{op}} \operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} ). \]