Variant 9.3.4.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a commutative diagram
\[ \xymatrix { Y_{\pm } \ar [r] \ar [d] & Y_{+} \ar [d] \ar [ddr] & \\ Y_{-} \ar [r] \ar [drr] & X_0 \ar [dr]^{u} & \\ & & X } \]
in $\operatorname{\mathcal{C}}$. If $u$ is a monomorphism, then the upper left square is a pullback diagram if and only if the outer square is a pullback diagram. This follows from Corollary 9.3.3.8 (combined with the characterization of pullback diagrams supplied by Proposition 7.6.2.14).