Remark 6.3.1.15 (038T)

Remark 6.3.1.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to a collection of edges $W$, and let $U: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories. Then, for every diagram $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, precomposition with $F$ induces a fully faithful functor

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{D}}, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \overline{\operatorname{\mathcal{E}}} ), \]

whose essential image is spanned by those functors $G: \operatorname{\mathcal{C}}\rightarrow \overline{\operatorname{\mathcal{E}}}$ which carry each edge of $W$ to an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}$. This follows by applying Corollary 4.5.2.32 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \overline{\operatorname{\mathcal{E}}} ) \ar [r]^{\circ F} \ar [d]^{ U \circ } & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \overline{\operatorname{\mathcal{E}}} ) \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}). } \]