Kerodon

Example 7.7.3.7. Let $\operatorname{\mathcal{C}}$ be a category which admits finite products, and let $e: M \times X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $e$ exhibits $M$ as an exponential of $Y$ by $X$ (in the sense of Definition 7.7.3.1). if and only if it exhibits $M$ as an exponential of $Y$ by $X$ in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.7.3.4). In particular, the category $\operatorname{\mathcal{C}}$ is cartesian closed if and only if the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is cartesian closed.