Remark 9.1.5.5. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category and let $K$ be a small simplicial set. It follows from the proof of Variant 9.1.5.4 that the diagonal functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is right cofinal if and only if the following condition is satisfied:
- $(\ast _0)$
If $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ can be written as an $K^{\operatorname{op}}$-indexed limit of corepresentable functors, then the colimit $\varinjlim (\mathscr {F} )$ is contractible.
Compare with condition $(\ast )$ of Proposition 7.7.7.3.