Proposition 7.1.7.6. Suppose we are given a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{F} \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}. } \]
where the vertical maps are inner fibrations. Let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ be a morphism of simplicial sets. If $F \circ \overline{q}$ is an $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$, then $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}'$.