Remark 7.7.2.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets whose fibers are essentially small $\infty $-categories, and let
be a contravariant transport representation for $U$ (Definition 8.6.8.1). If $K$ is a small simplicial set equipped with a morphism $F: K \rightarrow \operatorname{\mathcal{C}}$, then $\operatorname{Fun}^{\operatorname{ACart}}_{/\operatorname{\mathcal{C}}}(K, \operatorname{\mathcal{E}})$ can be identified with a limit of the composite diagram $K^{\operatorname{op}} \xrightarrow { F^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}} \operatorname{\mathcal{QC}}$ (Proposition 8.6.8.11). More generally, if $\kappa $ is an uncountable cardinal for which the cartesian fibration $U$ is essentially $\kappa $-small and the $\infty $-category $\operatorname{Fun}^{\operatorname{ACart}}_{/\operatorname{\mathcal{C}}}(K, \operatorname{\mathcal{E}})$ is essentially $\kappa $-small, then $\operatorname{Fun}^{\operatorname{ACart}}_{/\operatorname{\mathcal{C}}}(K, \operatorname{\mathcal{E}})$ can be realized as the limit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \circ F^{\operatorname{op}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$; see Proposition 8.6.8.11.