Kerodon

Warning 11.10.7.6. In the situation of Definition 11.10.7.1, we will often abuse terminology by referring to the morphism $f: X \rightarrow Y$ in the slice category $(\operatorname{Set_{\Delta }})_{/S}$ as a covariant equivalence (contravariant equivalence) if it is a covariant equivalence relative to $S$. Beware that there is some danger of confusion, because the notion of covariant equivalence depends on $S$. For example, the inclusion map $\{ 1\} \hookrightarrow \Delta ^1$ is not a covariant equivalence relative to $\Delta ^1$ (Warning 11.10.7.5), but is a covariant equivalence relative to the final object $\Delta ^{0} \in \operatorname{Set_{\Delta }}$ (Example 11.10.7.4).