# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 4.5.2.25. 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.

Proof. Apply Corollary 4.5.2.24 in the special case $\operatorname{\mathcal{C}}_1 = \{ C\}$ and $\operatorname{\mathcal{D}}_1 = \{ D\}$. $\square$