$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.9.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories and let $K$ be a simplicial set. The following conditions are equivalent:
- $(1)$
For every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ admits $K$-indexed colimits. Moreover, for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ preserves $K$-indexed colimits.
- $(2)$
Every lifting problem
7.45
\begin{equation} \begin{gathered}\label{equation:all-relative-colimits} \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{f} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}
admits a solution $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which is a $U$-colimit diagram.
Proof.
The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.3.9.5. Conversely, suppose that $(2)$ is satisfied. For each object $D \in \operatorname{\mathcal{D}}$, condition $(2)$ guarantees that every diagram $f: K \rightarrow \operatorname{\mathcal{C}}_{D}$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ which is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$. In particular, $\overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}_{D}$ (Corollary 7.1.5.20) having the property that for every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the composition $e_{!} \circ \overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}_{D'}$ (Proposition 7.3.9.2). To complete the proof, we observe that if $\overline{f}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D}$ is any other colimit diagram satisfying $\overline{f}'|_{K} = f$, then $\overline{f}'$ is isomorphic to $\overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_{D} )$, so that $e_{!} \circ \overline{f}'$ is also a colimit diagram in $\operatorname{\mathcal{C}}_{D'}$ (Corollary 7.1.2.14).
$\square$