Remark 7.7.2.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a diagram carrying the cone point of $K^{\triangleright }$ to an object $C \in \operatorname{\mathcal{C}}$, and let $\widetilde{C}$ be the object of $\operatorname{\mathcal{C}}_{/C}$ corresponding to the identity morphism $\operatorname{id}_{C}: C \rightarrow C$. Let $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/C}$ be as in Definition 7.7.2.24 and let $\widetilde{F}_0 = \widetilde{F}|_{K}$, so that $\widetilde{F}$ determines a natural transformation $\beta $ from $\widetilde{F}_0$ to the constant diagram taking the value $\widetilde{C}$. The following conditions are equivalent:
- $(1)$
The morphism $F$ is a weak colimit diagram: that is, $\widetilde{F}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$.
- $(2)$
The natural transformation $\beta $ exhibits $\widetilde{C}$ as a colimit of the diagram $\widetilde{F}_0$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$.
- $(3)$
There exists a final object $X \in \operatorname{\mathcal{C}}_{/C}$ and a natural transformation $\widetilde{F}_0 \rightarrow \underline{X}$ which exhibits $X$ as a colimit of $\widetilde{F}_0$.
- $(4)$
For any final object $X \in \operatorname{\mathcal{C}}_{/C}$, any natural transformation $\widetilde{F}_0 \rightarrow \underline{X}$ exhibits $X$ as a colimit of $\widetilde{F}_0$.
The equivalence $(1) \Leftrightarrow (2)$ follows from Remark 7.1.3.6 and the implications $(2) \Rightarrow (3)$ and $(4) \Rightarrow (2)$ follow from the observation that $X = \widetilde{C}$ is a final object of $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22). The implication $(3) \Rightarrow (4)$ follows from the observation that if $X$ is a final object of $\operatorname{\mathcal{C}}_{/C}$, then the constant diagram $\underline{X}$ is a final object of the diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (Proposition 7.1.7.2).