Kerodon

Remark 7.7.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products and let $e: M \times X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $e$ exhibits $M$ as an exponential of $Y$ by $X$ (in the sense of Definition 7.7.3.4, then the homotopy class $[e]$ exhibits $M$ as an exponential of $Y$ by $X$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 7.7.3.1). In particular, if the $\infty $-category $\operatorname{\mathcal{C}}$ is cartesian closed, then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is cartesian closed. Beware that the converse is false in general.