$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.3.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let
9.24
\begin{equation} \begin{gathered}\label{equation:pullback-over-discrete} \xymatrix { X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \end{gathered} \end{equation}
be a pullback diagram in $\operatorname{\mathcal{C}}$. If the object $X$ is discrete, then (9.24) determines a pullback diagram in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
Proof of Proposition 9.3.2.8.
By virtue of Proposition 7.6.2.14, it will suffice to show that the the tautological map $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$ preserves products. Since $X$ is discrete, we can use Remark 9.3.2.7 to identify $F$ with the canonical map from $\operatorname{\mathcal{C}}_{/X}$ to (the nerve of) its homotopy category, which always preserves products (see Warning 7.6.1.12).
$\square$