Corollary 8.4.2.8. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $f: K \rightarrow \operatorname{\mathcal{C}}$. Assume that $K$ is $\mathrm{ecf}(\kappa )$-small, so that $F = h_{\bullet } \circ f$ admits a limit $\widehat{C} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })$. Then $\widehat{C}$ is a covariant transport representation for the left fibration $( \operatorname{\mathcal{C}}_{ / f } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since the functor $h_{\bullet }$ is fully faithful, it induces a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/f} \ar [d]^{U} \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })_{ / F} \ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r]^-{h_{\bullet }} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa } ). } \]
Our assumption that $\widehat{C}$ is a limit of $F$ guarantees that $U$ is equivalent to the right fibration $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \widehat{C} } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. The desired result now follows from Corollary 8.4.2.7. $\square$