Remark 7.6.6.38 (Base Change). Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. The following conditions are equivalent:
- $(1)$
The inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete.
- $(2)$
For every pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r] \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, } \]the inner fibration $U'$ is $\mathbb {K}$-cocomplete.
- $(3)$
For every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the inner fibration $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is $\mathbb {K}$-cocomplete.
In particular, if $U$ is a $\mathbb {K}$-cocomplete inner fibration, then every fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a $\mathbb {K}$-cocomplete $\infty $-category (Example 7.6.6.31). The converse holds if $U$ is locally cartesian (Example 7.6.6.30), but not in general (Example 7.6.6.32).