Remark 9.3.4.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\sigma $ denote the composite map
\[ \Delta ^1 \times \Delta ^1 \xrightarrow { (i,j) \mapsto ij } \Delta ^1 \xrightarrow {u} \operatorname{\mathcal{C}}, \]
which we depict as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [r]^-{\operatorname{id}} \ar [d]^{\operatorname{id}} & X_0 \ar [d]^{f} \\ X_0 \ar [r]^-{f} & X. } \]
Then $f$ is a monomorphism if and only if $\sigma $ is a pullback square in $\operatorname{\mathcal{C}}$. This follows by combining Remarks 9.3.4.16 and 9.3.2.16 (see Proposition 7.6.2.14).