Example 7.3.7.14. In the situation of Proposition 7.3.7.13, suppose that $\operatorname{\mathcal{B}}= \Delta ^0$, so that we can identify $F$ with a functor from $\operatorname{\mathcal{K}}^{\triangleleft } \times \operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. Let $X$ denote the cone point of $\operatorname{\mathcal{K}}^{\triangleright }$. For each object $C \in \operatorname{\mathcal{C}}$, the inclusion map $\operatorname{\mathcal{K}}\times \{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{K}}\times \operatorname{\mathcal{C}}_{/C}$ is right cofinal (see Corollary 7.2.1.19). Applying Corollary 7.2.2.2, we deduce that $F$ is $V$-left Kan extended from $\operatorname{\mathcal{K}}\times \operatorname{\mathcal{C}}$ if and only if the induced map
is a $V$-colimit diagram for each $C \in \operatorname{\mathcal{C}}$. Proposition 7.3.7.13 asserts that, if this condition is satisfied, then $F$ determines a $V'$-colimit diagram $\operatorname{\mathcal{K}}^{\triangleleft } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$; this is a special case of Corollary 7.1.7.14.