Remark 7.7.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category category which admits finite products and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. Then an object $M \in \operatorname{\mathcal{C}}$ is an exponential of $Y$ by $X$ if and only if it represents the functor
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \times X, Y ). \]
In particular, the object $M$ is unique up to isomorphism and depends functorially on $X$ and $Y$. To emphasize this dependence, we sometimes denote the object $M$ by $Y^{X}$.