Remark 6.3.1.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. For every $\infty $-category $\operatorname{\mathcal{E}}$, the canonical isomorphism $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}))$ restricts to an isomorphism of full subcategories
\[ \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) ). \]
This follows immediately from the criterion of Theorem 4.4.4.4.