Definition 7.6.6.28. Let $\mathbb {K}$ be a collection of simplicial sets. We will say that an inner fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if the following condition is satisfied:
- $(\ast )$
For each $K \in \mathbb {K}$ and each vertex $C \in \operatorname{\mathcal{C}}$, every diagram
\[ f: K \rightarrow \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ which is an edgewise $U$-colimit diagram, in the sense of Defininition 7.1.7.9.