$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 9.4.8.2. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa $-small, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $\kappa $-cocomplete. Then the projection map $V: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration. Moreover, for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor
\[ \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}}) = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \{ C' \} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) = \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}}) \]
preserves $\kappa $-small colimits.
Proof.
To show that $V$ is a cocartesian fibration, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex (Proposition 5.1.4.8). In this case, $U$ is exponentiable (Corollary 9.4.6.19), so the desired result follows from Variant 8.6.5.11. To prove the second assertion, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ with $C = 0$ and $C' = 1$. In this case, $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}_{C'}$ are full subcategories of $\operatorname{\mathcal{E}}$, and determine restriction functors $R: \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}})$ and $R': \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}})$. The covariant transport functor $e_{!}$ can then be identified with the composition $R' \circ R^{L}$, where $R^{L}$ is left adjoint to $R$ (given by left Kan extension along the inclusion $\operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$). Since the functor $R^{L}$ preserves all colimits (Corollary 7.1.4.22), we are reduced to showing that the functor $R'$ preserves $\kappa $-small colimits, which follows from the criterion of Proposition 7.1.7.3.
$\square$