Remark 7.4.2.16 (Existence of Covariant Transport Representations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets. In ยง5.6, we showed that $U$ admits a covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism (Theorem 5.6.0.2). In the special case where $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration of $\infty $-categories, we can give an alternative proof of this fact using the characterization of Proposition 7.4.2.13. By virtue of Corollary 7.4.2.15, it will suffice to show that the constant functor $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$ admits a left Kan extension along $U$. Using the criterion of Proposition 7.3.4.4, we are reduced to showing that for every object $C \in \operatorname{\mathcal{C}}$, the constant functor $\operatorname{\mathcal{E}}_{C} \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ admits a colimit, which follows from Example 7.1.2.10.
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