Definition 7.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ be a morphism of simplicial sets. We will say that $e$ exhibits $X$ as a power of $Y$ by $K$ if, for every object $W \in \operatorname{\mathcal{C}}$, the composition law $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$ of Construction 4.6.9.9 induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) )$.
We will say that $e$ exhibits $Y$ as a copower of $X$ by $K$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the composition law $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) )$.