Kerodon

Warning 9.1.4.6. The converse of Corollary 9.1.4.5 is false in general. For example, suppose that $\operatorname{\mathcal{K}}$ is the nerve of the linearly ordered set $\{ 0 < 1 < 2 < \cdots \} $, and let $L$ be a simplicial set having exactly one vertex. Then the diagonal map $\operatorname{\mathcal{K}}\rightarrow \operatorname{Fun}( L^{\operatorname{op}}, \operatorname{\mathcal{K}})$ is an isomorphism (in particular, it is right cofinal). But it is usually not true that sequential colimits commute with $L$-indexed limits in the $\infty $-category $\operatorname{\mathcal{S}}$.