Kerodon

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Remark 8.2.3.7 (Duality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $\mathscr {H}': \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be the functor obtained from $\mathscr {H}$ by transposing its arguments. If $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, then $\mathscr {H}'$ is a $\operatorname{Hom}$-functor for the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. More precisely, if $\alpha : \underline{ \Delta ^{0} }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, then it also exhibits $\mathscr {H}'$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (by means of the identification $\operatorname{Tw}(\operatorname{\mathcal{C}}) \simeq \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} )$ supplied by Remark 8.1.1.6).