Kerodon

Remark 6.3.1.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ be an $\infty $-category. Then the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is replete. That is, if $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ are isomorphic objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, then $F$ carries edges of $W$ to isomorphisms in $\operatorname{\mathcal{E}}$ if and only if $F'$ carries edges of $W$ to isomorphisms in $\operatorname{\mathcal{E}}$ (see Example 4.4.1.14).