Warning 7.7.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. We have now assigned two different meanings to the notation $Y^{X}$:
If $X$ is an object of $\operatorname{\mathcal{C}}$, then $Y^{X}$ can denote an exponential of $Y$ by $X$ (Definition 7.7.3.4), characterized by the existence of natural homotopy equivalences
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (-), Y^{X} ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (-) \times X, Y). \]If $X$ is a simplicial set, then $Y^{X}$ can denote a power of $Y$ by $X$ (Definition 7.1.2.1), characterized by the existence of natural homotopy equivalences
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (-), Y^{X} ) \xrightarrow {\sim } \operatorname{Fun}(X, \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (-), Y) ). \]
Note that these are distinct notions in general. However, they agree in the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ and $X$ is a Kan complex. Beware that they do not agree in the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$ and $X$ is an $\infty $-category which is not a Kan complex.