Kerodon

Definition 4.7.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. Suppose we are given a pair of diagrams $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$. An isomorphism of $f_0$ with $f_1$ relative to $A$ is an isomorphism $u: f_0 \rightarrow f_1$ in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ for which the image of $u$ in $\operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is an identity morphism. We say that $f_0$ is isomorphic to $f_1$ relative to $A$ if there exists an isomorphism of $f_0$ with $f_1$ relative to $A$.