Remark 7.7.3.9. Let $\operatorname{\mathcal{C}}$ be a cartesian closed category. Suppose that $\operatorname{\mathcal{C}}$ is equipped with a simplicial enrichment having the following properties:
- $(1)$
The simplicial category $\operatorname{\mathcal{C}}$ is locally Kan: that is, for every pair of objects $C,X \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet }$ is a Kan complex.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$ and every finite collection of objects $\{ X_ i \} _{i \in I}$ having product $X = \prod _{i \in I}$, the canonical map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X)_{\bullet } \rightarrow \prod _{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, X_ i)_{\bullet } \]is an isomorphism of simplicial sets (rather than merely bijective on vertices).
- $(3)$
Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: Y^{X} \times X \rightarrow Y$ exhibit $Y^{X}$ as an exponential of $Y$ by $X$. Then, for every object $C \in \operatorname{\mathcal{C}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y^{X})_{\bullet } \xrightarrow { \times X} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C \times X, Y^{X} \times X)_{\bullet } \xrightarrow { e \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C \times X, Y)_{\bullet } \]is an isomorphism of simplicial sets (rather than merely bijective on vertices).
Then the homotopy coherent nerve $\operatorname{\mathcal{D}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is a cartesian closed $\infty $-category. Condition $(1)$ guarantees that $\operatorname{\mathcal{D}}$ is an $\infty $-category (Theorem 2.4.5.1), condition $(2)$ guarantees that the inclusion map $\iota : \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) = \operatorname{\mathcal{D}}$ preserves finite products (Example 7.6.1.15), and condition $(3)$ guarantees that $\iota $ carries exponentials to exponentials (see Theorem 4.6.8.5.