Proposition 7.6.2.29 (Rewriting Limits as Pullbacks). Suppose we are given a categorical pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ K \ar [r] \ar [d] & K_0 \ar [d] \\ K_1 \ar [r] & K_{01}. } \]

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, $K_0$-indexed limits, and $K_1$-indexed limits, then it also admits $K_{01}$-indexed limits. Moreover, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves pullback squares, $K$-indexed limits, $K_0$-indexed limits, and $K_1$-indexed limits, then $F$ also preserves $K$-indexed limits.