Remark 7.3.5.4. In the situation of Corollary 7.3.5.3, suppose we are given a functor $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and a natural trnasformation $\beta : F_0 \rightarrow F \circ U$. Then $\beta $ exhibits $F$ as a left Kan extension of $F$ along $U$ if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the induced natural transformation $\beta _{C}: F_0 |_{ \mathscr {G}(C) } \rightarrow \underline{ F(C) }$ exhibits $F(C)$ as a colimit of the diagram $F_0 |_{ \mathscr {G}(C)}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$