Proposition 4.5.2.18 (Transitivity). Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \ar [r] & \operatorname{\mathcal{C}}'' \ar [d] \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}'', } \]
where the square on the right is a categorical pullback. Then the square on the left is a categorical pullback if and only if the outer rectangle is a categorical pullback.