Kerodon

Warning 4.8.5.16. In the case $n = 0$, assertion $(2)$ of Proposition 4.8.5.15 is not necessarily true. For example, let $\operatorname{\mathcal{D}}$ be an $\infty $-category with the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{D}}$, there exists a morphism from $X$ to $Y$. Then the projection map $G: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ is categorically $1$-connective. Any choice of object $D \in \operatorname{\mathcal{D}}$ determines a functor $F: \{ D\} \hookrightarrow \operatorname{\mathcal{D}}$ with the property that $G \circ F$ is an isomorphism (and therefore categorically $0$-connective). However, the functor $F$ need not be essentially surjective.