Remark 9.4.8.5. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty $-categories which is essentially $\kappa $-small. Suppose we are given a commutative diagram of simplicial sets
which we identify with a morphism $H: \operatorname{\mathcal{D}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. Using the criterion of Variant 8.6.5.11, we see that $h$ factors through $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ if and only if $H$ satisfies the following condition:
- $(\ast )$
Let $X$ be a vertex of $\operatorname{\mathcal{D}}$ having image $C \in \operatorname{\mathcal{C}}$, let $e$ be an edge of $\{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$, and let $H_{e}$ denote the composition
\[ \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {e \times \operatorname{id}} \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {H} \operatorname{\mathcal{S}}^{< \kappa }. \]Then the functor $H_{e}$ is left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
Moreover, it suffice to verify condition $(\ast )$ under the additional assumption that $e$ belongs to the image of the equivalence $\operatorname{\mathcal{C}}_{C/} \rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (see Proposition 8.1.2.9).