Remark 9.1.4.8. Let $\operatorname{\mathcal{K}}$ be a small $\infty $-category and let $L$ be a small simplicial set. It follows from the proof of Variant 9.1.4.7 that the diagonal functor $\operatorname{\mathcal{K}}\rightarrow \operatorname{Fun}( L^{\operatorname{op}}, \operatorname{\mathcal{K}})$ is right cofinal if and only if the following condition is satisfied:
- $(\ast ')$
If $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{S}})$ can be written as an $L$-indexed limit of corepresentable functors, then the colimit $\varinjlim (\mathscr {F} )$ is contractible.
Compare with condition $(\ast )$ of Proposition 9.1.4.3.