Example 7.1.2.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be small $\infty $-categories, let $e_0: K \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ be a morphism of small Kan complexes, and let $e$ denote the composition of $e_0$ with the homotopy equivalence $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Remark 5.5.4.5. Combining Propositions 7.1.2.8 and 4.4.3.22, we obtain the following:
The morphism $e$ exhibits $\operatorname{\mathcal{C}}$ as a power of $\operatorname{\mathcal{D}}$ by $K$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if the induced map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories.
The morphism $e$ exhibits $\operatorname{\mathcal{C}}$ as a copower of $\operatorname{\mathcal{D}}$ by $K$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if the induced map $K \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.