Notation 7.6.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $K$ be a simplicial set. Suppose that there exists an object $X \in \operatorname{\mathcal{C}}$ and a morphism $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ which exhibits $X$ as a power of $Y$ by $K$. In this case, the object $X$ is uniquely determined up to isomorphism. To emphasize this uniqueness, we will sometimes denote the object $X$ by $Y^{K}$.
Similarly, if there exists an object $Z \in \operatorname{\mathcal{C}}$ and a morphism $e: K \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$ which exhibits $Z$ as a tensor product of $Y$ by $K$, then $Z$ is uniquely determined up to isomorphism. We will sometimes emphasize this dependence by denoting the object $Z$ by $K \otimes Y$.