Remark 7.7.2.20. Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits, and suppose we are given a diagram $\overline{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, carrying the cone point of $K^{\triangleright }$ to an object $C \in \operatorname{\mathcal{C}}$. Set $F = \overline{F}|_{K}$, so that $\overline{F}$ determines a natural transformation $\beta : F \rightarrow \underline{C}$. By virtue of Proposition 7.7.1.13 and Remark 7.1.3.6, $\overline{F}$ is a universal colimit diagram if and only if the following condition is satisfied:
- $(a)$
Let $u: C' \rightarrow C$ be a morphism in $\operatorname{\mathcal{C}}$ and let $\sigma :$
\[ \xymatrix { F' \ar [d]^{ \gamma } \ar [r]^{ \beta ' } & \underline{C}' \ar [d]^{ \underline{u} } \\ F \ar [r]^{ \beta } & \underline{C} } \]be a diagram in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. If $\sigma $ is a levelwise pullback square, then $\beta '$ exhibits $C'$ as a colimit of $F'$.
Suppose this condition is satisfied. Corollary 7.7.2.19 then asserts that $\overline{F}$ is a strongly universal colimit diagram if and only if it satisfies the following converse of $(a)$:
- $(b)$
For every diagram $\sigma $ as above, if $\beta '$ exhibits $C'$ as a colimit of $F'$ and $\gamma $ is cartesian, then $\sigma $ is a levelwise pullback square.