Variant 7.6.6.29. Some special cases of Definition 7.6.6.28 are sufficiently important to deserve their own terminology:
If $\kappa $ is an infinite cardinal, we say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it is $\mathbb {K}$-cocomplete, where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets.
We say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is finitely cocomplete if it is $\mathbb {K}$-cocomplete, where $\mathbb {K}$ is the collection of all finite simplicial sets.
We say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is cocomplete if it is $\mathbb {K}$-cocomplete, where $\mathbb {K}$ is the collection of all small simplicial sets.
Note that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is finitely cocomplete if and only if it is $\aleph _0$-cocomplete. Following the convention of Remark 4.9.0.4, $U$ is cocomplete if and only if it is $ \Omega $-cocomplete, where $ \Omega $ is some fixed strongly inaccessible cardinal.