Theorem 7.7.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $F$ is a universal colimit diagram (Definition 7.7.1.15), then $F$ is a descent diagram for the cartesian fibration $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. The converse holds if $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits or if $\operatorname{\mathcal{C}}$ has a final object.
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Proof of Theorem 7.7.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a universal colimit diagram. Then, for every cartesian natural transformation $\beta : F' \rightarrow F$, the morphism $F': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram, and therefore a weak colimit diagram (Proposition 7.7.2.26). Applying Corollary 7.7.2.31, we deduce that $F$ is a descent diagram for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. The converse holds whenever every weak colimit diagram $F': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram. By virtue of Proposition 7.7.2.26, this condition is satisfied if $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits or finite products (which follows from the existence of a final object of $\operatorname{\mathcal{C}}$, since $\operatorname{\mathcal{C}}$ admits pullbacks: see Corollary 7.6.2.30). $\square$