Definition 7.7.3.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category category which admits finite products and let $X$, $Y$, and $M$ be objects of $\operatorname{\mathcal{C}}$. We say that a morphism $e: M \times X \rightarrow Y$ exhibits $M$ as an exponential of $Y$ by $X$ if, for every object $C \in \operatorname{\mathcal{C}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, M ) \xrightarrow {\times X} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \times X, M \times X) \xrightarrow { [e] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \times X, Y) \]
is a homotopy equivalence of Kan complexes.
We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is cartesian closed if it admits finite products and, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $M \in \operatorname{\mathcal{C}}$ and a morphism $e: M \times X \rightarrow Y$ which exhibits $M$ as an exponential of $Y$ by $X$.