Remark 7.6.2.13 (Cofinality and Kan Extensions). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $\delta $ is left cofinal.
- $(2)$
The identity transformation $\operatorname{id}: \underline{ \Delta ^0 }_{K} \rightarrow \underline{ \Delta ^0}_{\operatorname{\mathcal{C}}} \circ \delta $ exhibits the constant functor $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^0 }_{K}: K \rightarrow \operatorname{\mathcal{S}}$ along $\delta $.
By virtue of Theorem 7.2.3.1 and Example 7.6.2.12, both conditions are equivalent to the requirement that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $K_{/C} = K \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is weakly contractible.