Warning 9.1.5.3. The converse of Proposition 9.1.5.2 is false in general. For example, suppose that $\operatorname{\mathcal{C}}$ is the nerve of the linearly ordered set $\{ 0 < 1 < 2 < \cdots \} $, and let $K$ be a connected Kan complex. Then the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is an isomorphism (in particular, it is right cofinal). But it is generally not true that sequential colimits commute with $K^{\operatorname{op}}$-indexed limits in the $\infty $-category $\operatorname{\mathcal{S}}$.
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