Kerodon

Remark 4.7.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams, and let $A \subseteq B$ be a simplicial subset. If $f_0$ and $f_1$ are isomorphic relative to $A$ (in the sense of Definition 4.7.6.1), then they are homotopic relative to $A$ (in the sense of Definition 3.2.1.1). The converse holds if the restriction functor $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is conservative. In particular, the converse holds if $\operatorname{\mathcal{C}}$ is a Kan complex, or if $A$ contains every vertex of $B$.