Corollary 8.3.1.8. Let $\kappa $ be an uncountable cardinal, let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally $\kappa $-small $\infty $-categories, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ and $\mathscr {G}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be functors, and let $\beta : \mathscr {F} \rightarrow \mathscr {G} \circ T$ be a natural transformation of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{S}}$. Fix an object $C \in \operatorname{\mathcal{C}}$ and a vertex $\eta \in \mathscr {F}(C)$ which exhibits the functor $\mathscr {F}$ as corepresented by $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The natural transformation $\beta $ carries $\eta $ to a vertex of $\mathscr {G}( T(C) )$ which exhibits the functor $\mathscr {G}$ as corepresented by the object $T(C) \in \operatorname{\mathcal{D}}$.
- $(2)$
The natural transformation $\beta $ exhibits $\mathscr {G}$ as a left Kan extension of $\mathscr {F}$ along the functor $T$ (see Variant 7.3.1.5).