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Proposition 4.5.2.20. Suppose we are given a commutative diagram of $\infty $-categories

4.26
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}

where $U$ is an isofibration. Then (4.26) is a categorical pullback square if and only if the induced map $\theta : \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories.

Proof. For every simplicial set $X$, Corollary 4.4.5.7 guarantees that the induced map $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$ is a Kan fibration. Combining Proposition 4.5.2.12 with Example 3.4.1.3, we see that (4.26) is a categorical pullback square if and only if it induces a homotopy equivalence

\[ \rho _{X}: \operatorname{Fun}( X, \operatorname{\mathcal{C}}')^{\simeq } \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})^{\simeq } \times _{ \operatorname{Fun}( X, \operatorname{\mathcal{D}})^{\simeq } } \operatorname{Fun}(X, \operatorname{\mathcal{D}}')^{\simeq }, \]

for every simplicial set $X$. Using Corollary 4.4.3.18, we can identify $\rho _ X$ with the map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}')^{\simeq } \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}' )^{\simeq }$ given by postcomposition with $\theta $. The desired result now follows from the criterion of Proposition 4.5.1.22. $\square$