Remark 9.3.4.31 (Pullbacks of Monomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a commutative diagram
9.27
\begin{equation} \begin{gathered}\label{equation:pullback-of-monomorphism} \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^{i} \ar [r] & Y_0 \ar [d]^{j} \\ X \ar [r]^-{f} & Y, } \end{gathered} \end{equation}
where $j: Y_0 \rightarrow Y$ is a monomorphism. Then (9.27) is a pullback square if and only if the following conditions are satisfied:
The morphism $i: X_0 \rightarrow X$ is also a monomorphism.
The diagram (9.27) determines a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. That is, a morphism $g: C \rightarrow X_0$ factors (up to homotopy) through $i$ if and only if the $f \circ g$ factors (up to homotopy) through $j$.
In particular, the collection of monomorphisms in $\operatorname{\mathcal{C}}$ is closed under pullbacks.