Remark 7.7.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram which has a universal colimit. Then, if $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is any colimit diagram extending $\mathscr {F}$, then $\overline{\mathscr {F}}$ is a universal colimit diagram. This follows from the observation that $\overline{\mathscr {F}}$ is unique up to isomorphism.
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