Kerodon

Warning 9.1.4.20. In the statement of Corollary 9.1.4.19, the hypothesis that $\operatorname{\mathcal{C}}$ is filtered cannot be omitted. For example, let $\operatorname{\mathcal{C}}$ denote the $\infty $-category given by the left cone $( \operatorname{\partial \Delta }^1 )^{\triangleleft }$. The inclusion map $\operatorname{\partial \Delta }^1 \hookrightarrow \Delta ^1$ then extends uniquely to a functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^1$, carrying the cone point of $\operatorname{\mathcal{C}}$ to the vertex $0 \in \Delta ^1$. The functor $F$ is right cofinal, but the induced map $F_{0/}: \operatorname{\mathcal{C}}_{0/} \rightarrow ( \Delta ^1 )_{0/} \simeq \Delta ^1$ is not.