Remark 8.6.6.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then the relative Yoneda embedding $T$ of Notation 8.6.6.13 fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} & \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d] \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \]
Consequently, for each vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $T$ to $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ \operatorname{id}_{C} \} $ can be regarded as a functor
\[ T_{C}: \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{S}}^{< \kappa } ), \]
which is a contravariant Yoneda embedding for the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, in the sense of Definition 8.3.3.9. See Remark 8.6.6.12.