Corollary 4.5.2.31. Suppose we are given a categorical pullback square of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [d]^-{U} \ar [r] & \widetilde{\operatorname{\mathcal{D}}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]
where $U$ and $V$ are isofibrations. Let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = F(C)$. Then the induced map
\[ \widetilde{\operatorname{\mathcal{C}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} = \widetilde{\operatorname{\mathcal{D}}}_{D} \]
is an equivalence of $\infty $-categories.