Example 7.6.2.11. Let $X$ and $Y$ be essentially small Kan complexes, let $e_0: K \rightarrow \operatorname{Fun}(X,Y)$ be a morphism of simplicial sets, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(X,Y)$ denote the composition of $e_0$ with the homotopy equivalence $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(X,Y)$ of Remark 5.5.1.5. Then:
The morphism $e$ exhibits $X$ as a power of $Y$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}$ and only the induced map $X \rightarrow \operatorname{Fun}(K,Y)$ is a homotopy equivalence of Kan complexes.
The morphism $e$ exhibits $Y$ as a tensor product of $X$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if the induced map $K \times X \rightarrow Y$ is a weak homotopy equivalence of simplicial sets.