Remark 9.1.1.11. In the situation of Remark 9.1.1.10, suppose that $\kappa $ is uncountable. If $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, then it satisfies the following a priori stronger versions of conditions $(2)$, $(3)$, and $(4)$:
- $(2')$
For every essentially $\kappa $-small simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty.
- $(3')$
For every essentially $\kappa $-small simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is nonempty.
- $(4')$
For every essentially $\kappa $-small simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, there exists a morphism $f \rightarrow f'$ in the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$, where $f': K \rightarrow \operatorname{\mathcal{C}}$ is a constant diagram.
The implication $(4) \Rightarrow (4')$ follows by choosing a categorical equivalence $K \rightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is a $\kappa $-small $\infty $-category. The equivalence $(3') \Leftrightarrow (4')$ is immediate, and the equivalence $(2') \Leftrightarrow (3')$ follows from Theorem 4.6.4.17.