Corollary 4.4.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms. Then the restriction map
\[ \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\quad \quad (f: X \rightarrow Y) \mapsto (X,Y) \]
is an isofibration of $\infty $-categories.