Example 7.6.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$, and suppose we are given a collection of morphisms $\{ f_ j: X \rightarrow Y \} _{j \in J}$ indexed by a set $J$. If we abuse notation by identifying $J$ with the corresponding discrete simplicial set, then the collection $\{ f_ j \} _{j \in J}$ can be identified with a map $e: J \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. In this case:
The morphism $e$ exhibits $X$ as a power of $Y$ by $J$ (in the sense of Definition 7.1.2.1) if and only if the collection $\{ f_ j \} _{j \in J}$ exhibits $X$ as a product of the collection $\{ Y \} _{j \in J}$ (in the sense of Definition 7.6.1.3). Stated more informally, we have a canonical isomorphism $Y^{J} \simeq {\prod }_{j \in J} Y$ (provided that either side is defined).
The morphism $e$ exhibits $Y$ as a copower of $X$ by $J$ (in the sense of Definition 7.1.2.1) if and only if the collection $\{ f_ j \} _{j \in J}$ exhibits $Y$ as a coproduct of the collection $\{ X \} _{j \in J}$ (in the sense of Definition 7.6.1.3). Stated more informally, we have a canonical isomorphism $J \otimes X \simeq {\coprod }_{j \in J} X$ (provided that either side is defined).