Remark 7.7.2.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets, and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism carrying the cone point to a vertex $C \in \operatorname{\mathcal{C}}$. For every vertex $x \in K$, the restriction of $F$ to $\{ x\} ^{\triangleright }$ determines an edge $e_{x}: F(x) \rightarrow C$ of $\operatorname{\mathcal{C}}$, which has an associated contravariant transport functor $e_{x}^{\ast }: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{ F(x) }$ (Definition 5.2.2.15). Using Remark 7.7.2.3 and Example 7.7.2.5, we can identify the restriction map $\theta $ appearing in Definition 7.7.2.6 with a comparison functor $\operatorname{\mathcal{E}}_{C} \rightarrow \varprojlim _{x \in K}( \operatorname{\mathcal{E}}_{F(x) } )$, given informally by the construction $(E in \operatorname{\mathcal{E}}_{C} ) \mapsto \{ e_{x}^{\ast } E \in \operatorname{\mathcal{E}}_{F(x)} \} _{x \in K}$. The morphism $F$ is an effective descent diagram for $\operatorname{\mathcal{C}}$ if and only if this comparison functor is an equivalence of $\infty $-categories. More precisely, if $\kappa $ is an uncountable cardinal for which $U$ is essentially $\kappa $-small, then $F$ is an effective descent diagram for $\operatorname{\mathcal{C}}$ if and only if the composition
is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$; here $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denotes a contravariant transport representation for the cartesian fibration $U$ (Proposition 8.6.8.13).