Corollary 8.5.4.26. Suppose we are given a categorical pullback diagram of $\infty $-categories
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\begin{equation} \begin{gathered}\label{equation:pullback-idempotent-complete} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^{ F_1 } & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then $\operatorname{\mathcal{C}}_{01}$ is idempotent complete.