Remark 7.1.2.4 (Duality). In the situation of Definition 7.1.2.1, the morphism $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ exhibits $X$ as a power of $Y$ by $K$ in the $\infty $-category $\operatorname{\mathcal{C}}$ if and only if the morphism
\[ e^{\operatorname{op}}: K^{\operatorname{op}} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}} \simeq \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }( Y, X ) \]
exhibits $X$ as a copower of $Y$ by $K^{\operatorname{op}}$ in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.