$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.6.6.40. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete. The following conditions are equivalent:
- $(1)$
The cocartesian fibration $U$ is $\mathbb {K}$-cocomplete, in the sense of Definition 7.6.6.28.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete and $U$ is $\mathbb {K}$-cocontinuous.
Proof.
We first show that $(1)$ implies $(2)$. Assume that $U$ is $\mathbb {K}$-cocomplete and suppose we are given a diagram $q: K \rightarrow \operatorname{\mathcal{E}}$ for some $K \in \mathbb {K}$. We wish to show that $q$ can be extended to a colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ which is preserved by the functor $U$. Set $q_0 = U \circ q$. Since $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete, we can extend $q_0$ to a colimit diagram $\overline{q}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. It follows from Example 7.6.6.37 that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ K \ar [d] \ar [r]^-{q} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ K^{\triangleright } \ar@ {-->}[ur]^{ \overline{q} } \ar [r]^-{ \overline{q}_0} & \operatorname{\mathcal{C}}} \]
admits a solution where $\overline{q}$ is a $U$-colimit diagram. Applying Corollary 7.1.6.12, we see that $\overline{q}$ is a colimit diagram which is preserved by the functor $U$.
We now prove the converse. Assume that $\operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete and that $U$ is $\mathbb {K}$-cocontinuous. Fix an object $C \in \operatorname{\mathcal{C}}$, a simplicial set $K \in \mathbb {K}$, and a diagram $q: K \rightarrow \operatorname{\mathcal{E}}_{C}$; we will complete the proof by showing that $q$ admits an extension $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ which is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ (Example 7.6.6.34). Using assumption $(2)$, we can extend $q$ to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ such that $\overline{q}_0 = U \circ \overline{q}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Let $\overline{q}'_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the constant functor taking the value $C$. Since $\overline{q}_0$ is a colimit diagram, we can choose a natural transformation $\alpha : \overline{q}_0 \rightarrow \overline{q}'_0$ which is the identity when restricted to $K$. Using Proposition 5.2.1.3, we can lift $\alpha $ to a $U$-cocartesian natural transformation $\widetilde{\alpha }: \overline{q} \rightarrow \overline{q}'$, where $\overline{q}': K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is an extension of $q$ and $\widetilde{\alpha }$ is the identity when restricted to $K$. Since $\overline{q}$ is a $U$-colimit diagram (Corollary 7.1.6.12), Proposition 7.3.9.3 guarantees that $\overline{q}'$ is also a $U$-colimit diagram.
$\square$